5-2 Medians and Altitudes of Triangles. , P is the centroid, PF = 6, and AD = 15. Find each measure.

Transcription

1 5-2 Medians Altitudes of Triangles In P the centroid PF = 6 AD = 15 Find each measure 10 3 INTERIOR DESIGN An interior designer creating a custom coffee table for a client The top of the table a glass triangle that needs to balance on a single support If the coordinates of the vertices of the triangle are at (3 6) (5 2) (7 10) at what point should the support be placed? 1 PC P the centroid of the triangle ACE by the Centroid Theorem We know that CF = PC + PF The triangle has a balance point at centroid Find the centroid of the triangular coffee table Use the centroid formula The centroid of the triangle Let the points be A(3 6) B(5 2) C(7 10) The midpoint D of 12 Note that 2 AP P the centroid of the triangle ACE a line that connects the vertex A D the midpoint of The dtance from D(6 6) to A(3 6) 6 3 or 3 units If P the centroid of the triangle ABC then by the Centroid Theorem So the centroid or 2 units to the right of A The coordinates of the centroid (P) are ( ) or (5 6) 10 3 INTERIOR DESIGN An interior designer creating a custom coffee table for a client The top of the table a glass triangle that needs to balance on a single support If the coordinates of the vertices of the triangle are at (3 6) (5 2) (7 10) at what point should the support be placed? (5 6) 4 COORDINATE GEOMETRY Find the coordinates of the orthocenter of triangle ABC with vertices A( 3 3) B( 1 7) C(3 3) The slope of So the slope of the altitude which perpendicular to Now the equation of the altitude from C to : Page 1

2 The slope of So the slope of the altitude which perpendicular to 5-2 Medians Altitudes of Triangles Now the equation of the altitude from C to : ( 1 5) In UJ = 9 VJ = 3 ZT = 18 Find each length 5 YJ Use the same method to find the equation of the altitude from A to That Solve the equations to find the intersection point of the altitudes Y the midpoint of a median of Similarly points T V are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid Theorem UJ So the coordinates of the orthocenter of 1 5) of UY ( 45 6 SJ V the midpoint of a median of Similarly points Y T are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid ( 1 5) Theorem SJ of SV In UJ = 9 VJ = 3 ZT = 18 Find each length Page 2

3 5-2 Medians Altitudes of Triangles 45 6 SJ 7 YU V the midpoint of a median of Similarly points Y T are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid Theorem SJ 6 of SV Y the midpoint of a median of Similarly points T V are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid Theorem UJ of UY 6 Now since YU the sum of YJ UJ we can add them to find YU 7 YU Y the midpoint of a median of Similarly points T V are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid Theorem UJ of UY SV V the midpoint of a median of Similarly points Y T are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid Theorem SJ of SV Now since YU the sum of YJ UJ we can add them to find YU Page 3 Therefore to find SV we can now add SJ VJ

4 5-2 Medians Altitudes of Triangles SV 9 9 JT V the midpoint of a median of Similarly points Y T are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid T the midpoint of a median of Similarly points Y V are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid Theorem SJ Theorem ZJ of ZT which we know ZT equals 18 from the given information To find JT subtract ZJ from ZT of SV Therefore to find SV we can now add SJ VJ JT T the midpoint of a median of Similarly points Y V are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid 10 ZJ T the midpoint of a median of Similarly points Y V are also midpoints of respectively so are also medians Therefore point J the centroid of according to the Centroid Theorem ZJ of ZT which we know ZT equals 18 from the given information Theorem ZJ of ZT which we know ZT equals 18 from the given information To find JT subtract ZJ from ZT 12 COORDINATE GEOMETRY Find the coordinates of the centroid of each triangle with the given vertices 11 A( 1 11) B(3 1) C(7 6) 6 The midpoint D of Page 4 Note that a line

5 (3 6) 5-2 Medians Altitudes of Triangles 12 COORDINATE GEOMETRY Find the coordinates of the centroid of each triangle with the given vertices 11 A( 1 11) B(3 1) C(7 6) 12 X(5 7) Y(9 3) Z(13 2) The midpoint D of The midpoint D of Note that a line that Note that a line that connects the vertex C D the midpoint of The dtance from D(1 6) to C(7 6) 7 1 or 6 units If P the centroid of the triangle ABC then So the centroid connects the vertex Z D the midpoint of The dtance from D(7 2) to Z(13 2) 13 7 or 6 units If P the centroid of the triangle XYZ then So the centroid or 4 units to the left of Z The coordinates of the centroid(p) are (13 4 2) or (9 2) or 4 units to the left of C The coordinates of the centroid (P) are (7 4 6) or (3 6) (9 2) 13 INTERIOR DESIGN Emilia made a collage with pictures of her friends She wants to hang the collage from the ceiling in her room so that it parallel to the ceiling A diagram of the collage shown in the graph at the right At what point should she place the string? (3 6) 12 X(5 7) Y(9 3) Z(13 2) The midpoint D of Note that a line that connects the vertex Z D the midpoint of The dtance from D(7 2) to Z(13 2) 13 7 or 6 units If P the centroid of the triangle XYZ then So the centroid or 4 units to the The triangle has a balance point at centroid Find the centroid of the picture Use the centroid formula left of Z The coordinates of the centroid(p) are (13 4 2) or (9 2) The centroid of the triangle Page 5

6 (3 4) 5-2 Medians Altitudes of Triangles (9 2) 13 INTERIOR DESIGN Emilia made a collage with pictures of her friends She wants to hang the collage from the ceiling in her room so that it parallel to the ceiling A diagram of the collage shown in the graph at the right At what point should she place the string? COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle with the given vertices 14 J(3 2) K(5 6) L(9 2) The slope of or So the slope of the altitude which perpendicular to Now the equation of the altitude from L to : The triangle has a balance point at centroid Find the centroid of the picture Use the centroid formula Use the same method to find the equation of the altitude from J to That Solve the equations to find the intersection point of the altitudes The centroid of the triangle (3 4) COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle with the given vertices 14 J(3 2) K(5 6) L(9 2) So the coordinates of the orthocenter of 1) The slope of or (5 So the slope of the altitude which perpendicular to Now the equation of the altitude from L to : Use the same method to find the equation of the altitude from J to That Solve the equations to find the intersection point of the altitudes (5 1) 15 R( 4 8) S( 1 5) T(5 5) Page 6

7 5-2 Medians (5 1) Altitudes of Triangles ( 4 4) 15 R( 4 8) S( 1 5) T(5 5) Identify each segment as a(n) altitude median or perpendicular bector 16 Refer to the image on page 340 The slope of or 1 So the slope of the altitude which perpendicular to the equation of the altitude from T to 1 Now : Use the same way to find the equation of the altitude from R to That Solve the equations to find the intersection point of the altitudes So the coordinates of the orthocenter of 4 4) ( an altitude here because it perpendicular to altitude 17 Refer to the image on page 340 ( 4 4) Identify each segment as a(n) altitude median or perpendicular bector 16 Refer to the image on page 340 a median because D the midpoint of the side opposite vertex B median 18 Refer to the image on page 340 perpendicular to as well as meets at the midpoint D Therefore an altitude median perpendicular bector an altitude here because it perpendicular to Page 7 perpendicular bector altitude median

8 a median because D the midpoint of the side opposite vertex B 5-2 Medians Altitudes of Triangles median 18 Refer to the image on page 340 a median because it connects point B to point D which the midpoint of median 20 CCSS SENSE-MAKING In the figure if J P L are the midpoints of respectively find x y z perpendicular to as well as meets at the midpoint D Therefore an altitude median perpendicular bector perpendicular bector altitude median By the Centroid Theorem 19 Refer to the image on page 340 a median because it connects point B to point D which the midpoint of median 20 CCSS SENSE-MAKING In the figure if J P L are the midpoints of respectively find x y z By the Centroid Theorem x = 475 y = 6 z = 1 Copy complete each statement for for medians centroid J 21 SL = x(jl) Page 8

9 5-2 Medians Altitudes of Triangles x = 475 y = 6 z = 1 Copy complete each statement for for medians centroid J 21 SL = x(jl) 23 JM = x(rj) 3 ALGEBRA Use the figure 22 JT = x(tk) 24 If m an altitude of 2 = 3x + 13 find m m 1 = 2x m 2 By the definition of altitude By the Centroid Theorem Substitute 14 for x in 23 JM = x(rj) Page 9 m 1 = 35 m 2 = 55

10 m 1 = 35 m 5-2 Medians Altitudes of Triangles 2 = Find the value of x if AC = 4x 3 DC = 2x + 9 m ECA = 15x + 2 a median of Is also an altitude of? Explain ALGEBRA Use the figure Given: AC = DC 24 If m an altitude of 2 = 3x + 13 find m m 1 = 2x m 2 By the definition of altitude Substitute 6 for x in m ECA Substitute 14 for x in not an altitude of because m ECA = 92 6; no; because m m 1 = 35 m 2 = Find the value of x if AC = 4x 3 DC = 2x + 9 m ECA = 15x + 2 a median of Is also an altitude of? Explain ECA = GAMES The game board shown shaped like an equilateral triangle has indentations for game pieces The game s objective to remove pegs by jumping over them until there only one peg left Copy the game board s outline determine which of the points of concurrency the blue peg represents: circumcenter incenter centroid or orthocenter Explain your reasoning Given: AC = DC Substitute 6 for x in m ECA not an altitude of esolutions 92 Manual - Powered by Cognero because m ECA = To determine each point of concurrency you must perform their corresponding constructions You may want to make a different tracing for each center so your lines arcs won't get confusingto determine if the blue peg a circumcenter construct the perpendicular bectors of each side To determine if the blue peg the incenter you need to construct the angle bectors of each side The centroid can be determined by constructing the midpoints of each Page 10 side connecting them to the opposite vertices Finally the orthocenter can be constructed by finding the altitudes from each vertex to the opposite side

11 not an altitude of because m ECA = side of the triangle so it also represents the perpendicular bector the median the altitude That means that the blue peg represents all of the centers including the circumcenter incenter centroid orthocenter Medians Altitudes of Triangles 6; no; because m ECA = GAMES The game board shown shaped like an equilateral triangle has indentations for game pieces The game s objective to remove pegs by jumping over them until there only one peg left Copy the game board s outline determine which of the points of concurrency the blue peg represents: circumcenter incenter centroid or orthocenter Explain your reasoning CCSS ARGUMENTS Use the given information to determine whether a perpendicular bector median /or an altitude of 27 To determine each point of concurrency you must perform their corresponding constructions You may want to make a different tracing for each center so your lines arcs won't get confusingto determine if the blue peg a circumcenter construct the perpendicular bectors of each side To determine if the blue peg the incenter you need to construct the angle bectors of each side The centroid can be determined by constructing the midpoints of each side connecting them to the opposite vertices Finally the orthocenter can be constructed by finding the altitudes from each vertex to the opposite side The blue peg represents all of the centers including the circumcenter incenter centroid orthocenter Circumcenter incenter centroid orthocenter; Sample answer: The angle bector of each angle also bects the opposite side perpendicular to the opposite side of the triangle so it also represents the perpendicular bector the median the altitude That means that the blue peg represents all of the centers including the circumcenter incenter centroid orthocenter CCSS ARGUMENTS Use the given information to determine whether a perpendicular bector median /or an altitude of an altitude by the definition of altitudewe don't know if it a perpendicular bector because it not evident that M the midpoint of altitude 28 we know that by CPCTC they are a linear pair then we know they are right angles Therefore the perpendicular bector median altitude of perpendicular bector median altitude 29 we know that midpoint of Therefore then M the the median of median we can prove by HL Therefore we know that by CPCTC making M the midpoint of Therefore the perpendicular bector Page 11 median altitude of

12 midpoint of Therefore the median of 5-2 Medians Altitudes of Triangles median 30 perpendicular bector median altitude we can prove by HL Therefore we know that by CPCTC making M the midpoint of Therefore the perpendicular bector median altitude of perpendicular bector median altitude PROOF Write a paragraph proof 31 Given: osceles bects Prove: a median Y When solving th proofs like these it helps to think backwards What do you need to do to prove that a median of the triangle? To be considered a median then it must be formed by a segment with one endpoint on a vertex of the triangle the other endpoint at the midpoint of the opposite side To prove that a median you must prove that W the midpoint of or that Use the given information to prove that the two triangles in the diagram are congruent then use as your CPCTC statement Given: Prove: osceles a median bects Y Proof: osceles By the definition of angle bector by the Reflexive Property So by SAS By CPCTC By the definition of a midpoint W the midpoint of By the definition of a median a median PROOF Write a paragraph proof 31 Given: osceles bects Prove: a median osceles a median bects esolutions Manual - Powered by Proof: Cognero osceles Given: Prove: osceles a median bects Y Proof: osceles By the definition of angle bector by the Reflexive Property So by SAS By CPCTC By the definition of a midpoint W the midpoint of By the definition of a median a median Given: Prove: osceles a median bects Y Proof: osceles definition of angle bector Property So by SAS CPCTC By the by the Reflexive By By the definition of a midpoint W the midpoint of By the definition of a a median Y By the definition of angle bector Y When solving th proofs like these it helps to think backwards What do you need to do to prove that a median of the triangle? To be considered a median then it must be formed by a segment with one endpoint on a vertex of the triangle the other endpoint at the midpoint of the opposite side To prove that a median you must prove that W the midpoint of or that Use the given information to prove that the two triangles in the diagram are congruent then use as your CPCTC statement median Given: Prove: of Therefore the perpendicular bector median altitude of PROOF Write an algebraic proof 32 Given: with medians Prove: by the Reflexive Page 12

13 CPCTC By the definition of a midpoint W the midpoint of 6 (Subtraction Property) 7 (Multiplication Property) By the definition of a median a median 5-2 Medians Altitudes of Triangles 8 PROOF Write an algebraic proof 32 Given: with medians (Divion Property) Given: Prove: with medians Prove: Proof: Statements (Reasons) 1 (Given) with medians Given: 2 Prove: Consider what algebraic relationships ext for the with medians (Centroid Theorem) 3 (Segment Addition Post) 4 (Substitution Property) 5 (Dtributive Property) medians of a triangle You know that according to the Centroid Theorem the centroid of a triangle lies the dtance from one vertex to the midpoint of the 6 (Subtraction Property) 7 (Multiplication Property) opposite side Using th you can set up an equation relating XP XR You also can write a statement 8 (Divion Property) relating XR to XP PR Notice how you have XR in your first two statements Think about how you can substitute one statement into the other to just leave XP XR in your equation Keep simplifying until you reach what you are trying to prove Proof: Statements (Reasons) 1 with medians (Given) 2 (Centroid Theorem) 3 (Segment Addition Postulate) 4 (Substitution Property) 5 (Dtributive Property) 6 (Subtraction Property) 7 (Multiplication Property) 8 a Using a different piece of patty paper for each equilateral triangle fold the triangles to find their circumcenter ( by making the perpendicular bectors) the incenter ( by making the angle bectors) centroid ( by constructing the medians) the orthocenter ( by making the altitudes of each side) (Divion Property) Page 13 Given: 33 MULTIPLE REPRESENTATIONS In th problem you will investigate the location of the points of concurrency for any equilateral triangle a CONCRETE Construct three different equilateral triangles on patty paper cut them out Fold each triangle to locate the circumcenter incenter centroid orthocenter b VERBAL Make a conjecture about the relationships among the four points of concurrency of any equilateral triangle c GRAPHICAL Position an equilateral triangle its circumcenter incenter centroid orthocenter on the coordinate plane using variable coordinates Determine the coordinates of each point of concurrency with medians

14 equilateral triangle fold the triangles to find their circumcenter ( by making the perpendicular bectors) the incenter ( by making the angle bectors) centroid ( by constructing the medians) 5-2 Medians Altitudes of Triangles the orthocenter ( by making the altitudes of each side) of the third vertex To locate the coordinate of the points of concurrency which are all concurrent in an equilateral triangle we need to use the Centroid Theorem to find the location that from the vertex to the midpoint of the opposite side Th would be the point that b Sample answer: The four points of concurrency of an equilateral triangle are all the same point c Th problem can be made easier by placing the triangle on the coordinate plane in such a way that one vertex placed at the origin one side of the triangle lies along the x-ax Additionally by choosing a coordinate for the vertex on the x-ax that divible by two you can easily compute the coordinate of the third vertex of the dtance from along the height or altitude of the triangle Th can be found as shown below: The y -value of the centroid Therefore the coordinate of the points of concurrency of the equilateral triangle would be Using the coordinates (0 0) (4a 0) for the two vertices on the x-ax we need to find the coordinates of the third vertex it located above the midpoint of the side across from it we know the x-value 2a Using the Converse of the Pythagorean Theorem we can set up an equation to find the y-value which also the height of the triangle The height divides the equilateral triangle into two parts with a height (h) half of the base of the equilateral triangle (2a) as its legs 4a as its hypotenuse Using the equation below we can find the height of the triangle: a b Sample answer: The four points of concurrency of an equilateral triangle are all the same point c Therefore the height of the triangle the y -value of the third vertex To locate the coordinate of the points of concurrency which are all concurrent in an equilateral triangle we need to use the Centroid Theorem to find the location that from the vertex to the midpoint of the opposite side Th would be the point that of the dtance from ALGEBRA In m 3y 2 LK = 5y 8 JMP = 3x 6 JK = Page 14

15 7 5-2 Medians Altitudes of Triangles PROOF Write a coordinate proof to prove the Centroid Theorem ALGEBRA In m 3y 2 LK = 5y 8 34 If an altitude of JMP = 3x 6 JK = find x an altitude of Given: medians Prove: The medians intersect at point P P two thirds of the dtance from each vertex to the midpoint of the opposite side (Hint: First find the equations of the lines containing the medians Then find the coordinates of point P show that all three medians intersect at point P Next use the Dtance Formula multiplication to show 35 Find LK if a median To begin th proof you will need to compute the slopes of the three medians Using these slopes determine the equation of each line containing these medians Set the equations of two medians equal to each other to determine the coordinates of point P the centroid You can then choose a different pair of medians to check if you get the same point P or you can plug point P into the third median to confirm that it does lie on that line as well Once you have proven that P the median of the triangle you can use its coordinates to find the lengths of the parts of each median using the Dtance formula Once th done you can substitute in the lengths of the medians a median JK = KL Substitute 3 for y in LK to verify that Proof: 7 Slope of ; PROOF Write a coordinate proof to prove the Centroid Theorem Slope of ; Slope of ; contained in the line contained in the line 36 Given: medians Page 15

16 Slope of ; 5-2 Medians Slope of Altitudes of Triangles ; contained in the line contained in the line contained in the line To find the coordinates of P find the intersection point of two medians Find y So the coordinates of P are (2b + 2a 2c) Now show that P on Thus the three medians intersect at the same point Find the lengths of using the Dtance Formula Show that the P two thirds of the dtance from the vertices to the midpoints Page 16

17 Find y 5-2 Medians Altitudes of Triangles Show that the P two thirds of the dtance from the vertices to the midpoints So the coordinates of P are (2b + 2a 2c) Now show that P on Thus the three medians intersect at the same point Find the lengths of using the Dtance Formula Thus Proof: Slope of ; Slope of ; Slope of ; contained in the line contained in the line the line contained in To find the coordinates of P find the intersection point of two medians Find y So the coordinates of P are (2b + 2a 2c) esolutions Cognero NowManual show- Powered that P byon Page 17

18 Thus 5-2 Medians Altitudes of Triangles 37 ERROR ANALYSIS Based on the figure at the right Luke says that Kareem dagrees Is either of them correct? Explain your reasoning Show that the P two thirds of the dtance from the vertices to the midpoints Sample answer: Kareem correct Luke has the segment lengths transposed The shorter segment 2/3 of the entire median not the other way around According to the Centroid Theorem Sample answer: Kareem correct According to the Centroid Theorem The segment lengths are transposed Thus 38 CCSS ARGUMENTS Determine whether the following statement true or false If true explain your reasoning If false provide a counterexample The orthocenter of a right triangle always located at the vertex of the right angle 37 ERROR ANALYSIS Based on the figure at the right Luke says that Kareem dagrees Is either of them correct? Explain your reasoning It true that "The orthocenter of a right triangle always located at the vertex of the right angle" Sample answer: As shown in the right triangle below the altitudes from the two nonright vertices (A B) will always be the legs of the triangle which intersect at the vertex that contains the right angle (C) The altitude to the hypotenuse of the triangle originates at the vertex so the three altitudes (the red rays) intersect there Therefore the vertex of a right triangle will always be the orthocenter Sample answer: Kareem correct Luke has the segment lengths transposed The shorter segment 2/3 of the entire median not the other way around According to the Centroid Theorem Sample answer: Kareem correct According to the Centroid Theorem lengths are transposed The segment Page 18

19 Sample answer: Kareem correct According to the Centroid Theorem The segment 5-2 Medians Altitudes of Triangles lengths are transposed 39 CHALLENGE has vertices A( 3 3) B(2 5) C(4 3) What are the coordinates of the centroid of? Explain the process you used to reach your conclusion 38 CCSS ARGUMENTS Determine whether the following statement true or false If true explain your reasoning If false provide a counterexample The orthocenter of a right triangle always located at the vertex of the right angle It true that "The orthocenter of a right triangle always located at the vertex of the right angle" Sample answer: As shown in the right triangle below the altitudes from the two nonright vertices (A B) will always be the legs of the triangle which intersect at the vertex that contains the right angle (C) The altitude to the hypotenuse of the triangle originates at the vertex so the three altitudes (the red rays) intersect there Therefore the vertex of a right triangle will always be the orthocenter The midpoint D of Note that that connects the vertex B D the midpoint of 39 CHALLENGE has vertices A( 3 3) B(2 5) C(4 3) What are the coordinates of the centroid of? Explain the process you used to reach your conclusion The slope of the line The equation of True; sample answer: In a right triangle the altitudes from the two nonright vertices will always be the legs of the triangle which intersect at the vertex that contains the right angle The altitude to the hypotenuse of the triangle originates at the vertex so the three altitudes intersect there Therefore the vertex of a right triangle will always be the orthocenter a line : Use the same method to find the equation of the line between point A the midpoint of That Solving the system of equations we get the So the centroid of the intersection point triangle ; Sample answer: I found the midpoint of used it to find the equation for the line that contains point B the midpoint of I also found the midpoint of the equation for the line between pointpage A 19 the midpoint of I solved the

20 ; Sample answer: I found the midpoint of used it to find the equation for the line that contains point B the midpoint of 5-2 Medians Altitudes of Triangles I also found the midpoint of the equation for the line between point A the midpoint of I solved the system of two equations for x y to get the coordinates of the centroid 40 WRITING IN MATH Compare contrast the perpendicular bectors medians altitudes of a triangle To consider th answer start by making a lt of the properties of each item Consider which ones must pass through the vertex which must meet at the midpoint of the opposite side as well as which ones must be perpendicular to the opposite side Try comparing contrasting two items at a time until all three have been compared Sample answer: The perpendicular bector the median pass through the midpoint on the side of the triangle but only the median always passes through the vertex opposite the side The perpendicular bector the altitude are both perpendicular to the side but do not necessarily pass through a common point on the side of the triangle The median the altitude both pass through the vertex but do not necessarily pass through a common point on the side of the triangle Sample answer: The perpendicular bector the median pass through a common point on the side of the triangle but only the median always passes through the vertex opposite the side The perpendicular bector the altitude are both perpendicular to the side but do not necessarily pass through a common point on the side of the triangle The median the altitude both pass through the vertex but do not necessarily pass through a common point on the side of the triangle 41 CHALLENGE In the figure segments 40 WRITING IN MATH Compare contrast the perpendicular bectors medians altitudes of a triangle To consider th answer start by making a lt of the properties of each item Consider which ones must pass through the vertex which must meet at the midpoint of the opposite side as well as which ones must be perpendicular to the opposite side Try comparing contrasting two items at a time until all three have been compared Sample answer: The perpendicular bector the median pass through the midpoint on the side of the triangle but only the median always passes through the vertex opposite the side The perpendicular bector the altitude are both perpendicular to the side but do not necessarily pass through a common point on the side of the triangle The median the altitude both pass through the vertex but do not necessarily pass through a common point on the side of the triangle Sample answer: The perpendicular bector the median pass through a common point on the side of the triangle but only the median always passes through the vertex opposite the side The perpendicular bector the altitude are both perpendicular to the side but do not necessarily pass through a common point on the side of the triangle The median the altitude both pass through the vertex but do not necessarily pass through a common point on the side of the triangle 41 CHALLENGE In the figure segments are medians of AB = 10 CE = 9 Find CA are medians the intersect at the centroid of the triangle Let P be the centroid the intersection point of the medians By the Centroid Theorem That CP = 6 And PE = 9 then two right triangles 6 = 3 Because are formed Using the Page 20 converse of the Pythagorean Theorem we can find the lengths of then as shown below:

21 perpendicular to the side but do not necessarily pass through a common point on the side of the triangle The median the altitude both pass through the vertex but do not necessarily pass through a 5-2 Medians Altitudes of Triangles common point on the side of the triangle 41 CHALLENGE In the figure segments are medians of AB = 10 CE = 9 Find CA c Draw a right triangle find the circumcenter centroid orthocenter d Make a conjecture about the relationships among the circumcenter centroid orthocenter a Make an acute triangle construct the centroid (medians) circumcenter (perpendicular bectors) orthocenter (altitudes)pay attention to how these three centers relate to each other are medians the intersect at the centroid of the triangle Let P be the centroid the intersection point of the medians By the Centroid Theorem That CP = 6 And PE = 9 then two right triangles 6 = 3 Because are formed Using the converse of the Pythagorean Theorem we can find the lengths of then as shown below: b Make an obtuse triangle construct the centroid (medians) circumcenter (perpendicular bectors) orthocenter (altitudes) Pay attention to how these three centers relate to each other In the right triangle APE In the right triangle APC 42 OPEN ENDED In th problem you will investigate the relationships among three points of concurrency in a triangle a Draw an acute triangle find the circumcenter centroid orthocenter b Draw an obtuse triangle find the circumcenter centroid orthocenter c Draw a right triangle find the circumcenter centroid orthocenter d Make a conjecture about the relationships among the circumcenter centroid orthocenter a Make an acute triangle construct the centroid (medians) circumcenter (perpendicular bectors) orthocenter (altitudes)pay attention to how these three centers relate to each other c Make a right triangle construct the centroid (medians) circumcenter (perpendicular bectors) orthocenter (altitudes) Pay attention to how these three centers relate to each other d Sample answer: The circumcenter centroid orthocenter are all collinear Page 21 a

22 5-2 Medians Altitudes of Triangles d Sample answer: The circumcenter centroid orthocenter are all collinear a b d Sample answer: The circumcenter centroid orthocenter are all collinear 43 WRITING IN MATH Use area to explain why the centroid of a triangle its center of gravity Then use th explanation to describe the location for the balancing point for a rectangle When considering th answer think about what the medians of a triangle do to the triangle each median divides the triangle into two smaller triangles of equal area the triangle can be balanced along any one of those lines To balance the triangle on one point you need to find the point where these three balance lines intersect The balancing point for a rectangle the intersection of the segments connecting the midpoints of the opposite sides since each segment connecting these midpoints of a pair of opposite sides divides the rectangle into two parts with equal area Sample answer: Each median divides the triangle into two smaller triangles of equal area so the triangle can be balanced along any one of those lines To balance the triangle on one point you need to find the point where these three balance lines intersect The balancing point for a rectangle the intersection of the segments connecting the midpoints of the opposite sides since each segment connecting these midpoints of a pair of opposite sides divides the rectangle into two parts with equal area c d Sample answer: The circumcenter centroid orthocenter are all collinear 43 WRITING IN MATH Use area to explain why the centroid of a triangle its center of gravity Then use th explanation to describe the location for the balancing point for a rectangle When considering th answer think about what the medians of a triangle do to the triangle each esolutions Manual - Powered by Cognero median divides the triangle into two smaller triangles of equal area the triangle can be balanced along any one of those lines To balance the triangle on one 44 In the figure below A B C D Which must be true? an altitude of an angle bector of a median of a perpendicular bector of We are given that therefore J the midpoint of We don't know if so all we can conclude that a median of The correct choice C C Page GRIDDED RESPONSE What the x-intercept of

23 balancing point for a rectangle the intersection of the segments connecting the midpoints of the opposite sides since each segment connecting these midpoints of a Altitudes pair of opposite sides divides the 5-2 Medians of Triangles rectangle into two parts with equal area 44 In the figure below A B C D Which must be true? an altitude of an angle bector of a median of a perpendicular bector of We are given that therefore J the midpoint of We don't know if so all we can conclude that a median of The correct choice C C 45 GRIDDED RESPONSE What the x-intercept of the graph of? Substitute y = 0 in the equation to find the xintercept 3 46 ALGEBRA Four students have volunteered to fold pamphlets for a local community action group Which student the fastest? F Deron G Neiva H Quinn J Sarah Find the folding speed of the students per second Neiva: pages per second Sarah: pages per second Quinn: pages per second Deron: pages per second Among these students Quinn the fastest student in folding pamphlets The correct choice H H 3 46 ALGEBRA Four students have volunteered to fold pamphlets for a local community action group Which student the fastest? 47 SAT/ACT 80 percent of 42 what percent of 16? A 240 B 210 C 150 D 50 E 30 Let x be the percent of 16 which equal to the 80 percent of 42 F Deron G Neiva H Quinn J Sarah Find the folding speed of the students per second Neiva: pages per second Page 23

24 The correct choice H Therefore 5-2 Medians Altitudes of Triangles H 47 SAT/ACT 80 percent of 42 what percent of 16? A 240 B 210 C 150 D 50 E 30 Let x be the percent of 16 which equal to the 80 percent of DF By the Converse of Perpendicular Bector Theorem By the Segment Addition Postulate Therefore 5 50 TQ The correct choice B B By the Perpendicular Bector Theorem RQ = TQ Find each measure 48 LM From the figure because bector of Therefore 12 the perpendicular DF Position label each triangle on the coordinate plane with hypotenuse ZY twice XY 51 right b units long By placing Y at the origin the legs of the right triangle are positioned along the x-ax y-ax b units long you can place point X along the y-ax make the coordinates (0b) twice the length of it lies along the x-ax Page 24 its coordinates would be (2b0)

25 5-2 Medians Altitudes of Triangles Position label each triangle on the coordinate plane 51 right with hypotenuse ZY twice XY b units long By placing Y at the origin the legs of the right triangle are positioned along the x-ax y-ax b units long you can place point X along the y-ax make the coordinates (0b) twice the length of it lies along the x-ax its coordinates would be (2b0) 52 osceles with base that b units long By placing Q at the origin the base of the osceles triangle can be positioned along the x-ax b units long you can place point R along the x-ax make the coordinates (b0) the vertex point T of the osceles triangle directly above the midpoint of the base its x-value would be half the length of its coordinates are Page 25

26 5-2 Medians Altitudes of Triangles Determine whether are parallel perpendicular or neither Graph each line to verify your answer 53 R(5 4) S(10 0) J(9 8) K(5 13) Slope of : 54 R(1 1) S(9 8) J( 6 1) K(2 8) Slope of : Slope of : the slopes are equal the lines are parallel Slope of : Neither parallel nor perpendicular Parallel neither Page 26

27 5-2 Medians Altitudes of Triangles 55 HIGHWAYS Near the city of Hopewell Virginia Route 10 runs perpendicular to Interstate 95 Interstate 295 Show that the angles at the intersections of Route 10 with Interstate 95 Interstate 295 are congruent 56 Given: Prove: Because Route 10 perpendicular to Interstate 295 a right angle The same true for Route 10 Interstate 95 therefore also a right angle both are right angles then they are congruent to each other because all right angles are congruent Therefore 1 congruent to 2 Th proof based on two main relationships - the Triangle Angle-Sum Theorem the Definition of a Linear Pair Based on the diagram write two equations using each of these relationships If two angles form a linear pair then we know that they are supplementary consequently add up to 180 degrees At th stage of the proof you can set the two equations equal to each other simplify to obtain the final conclusion of the proof Because the lines are perpendicular the angles formed are right angles All right angles are congruent Therefore 1 congruent to 2 PROOF Write a flow proof of the Exterior Angle Theorem Page 27

28 5-2 Medians Altitudes of Triangles Page 28