5-1 Perpendicular and Angle Bisectors
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1 5-1 Perpendicular and Angle Bisectors Equidistant Distance and Perpendicular Bisectors Theorem Hypothesis Conclusion Perpendicular Bisector Theorem Converse of the Perp. Bisector Theorem Locus Applying the Perpendicular Bisector Theorem and Its Converse 1a. Given that line l is the perpendicular bisector of DE and EG = 14.6, find DG. 1b. Given that DE = 20.8, DG = 36.4, and EG = 36.4, find EF. Distance and Angle Bisectors Theorem Hypothesis Conclusion Angle Bisector Theorem Converse of the Angle Bisector Theorem
2 Applying the Angle Bisector 2a. Given YW bisects XYZ and WZ = 3.05, find WX. 2b. Given that m WYZ = 63, XW = 5.7, and ZW = 5.7, find m XYZ. Parachute Application 3. Each pair of suspension lines on a parachute are the same length and are equally spaced form the center of the chute. S is equidistant from each pair of suspension lines. What can you conclude about QS? Writing Equations of Bisectors in the Coordinate Plane 4. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P(5,2) and Q(1, 4). Q: Where do math teachers shop? A: Deci-malls!
3 5-2 Bisectors of Triangles Concurrent Point of concurrency Circumcenter of the triangle Circumcenter Theorem Circumscribed Using Properties of Perpendicular Bisectors KZ, LZ, and MZ are perpendicular bisectors of GHJ. Use the diagram to find each length. 1a. GM 1b. GK 1c. JZ Finding the Circumcenter of a Triangle 2. Find the circumcenter of GOH with vertices G(0, 9), O(0,0), and H(8,0).
4 Incenter of a triangle Incenter Theorem Inscribed Using Properties of Angle Bisectors QX and RX are angle bisectors of PQR. Find each measure. 3a. the distance from X to PQ 3b. m PQX Community Application 4. A city plans to build a firefighters monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch. Mr. Graves: How many sides does a circle have? Student: Two inside and outside!
5 5-3 Medians and Altitudes of Triangles Median of a triangle Centroid of the triangle Centroid Theorem Using the Centroid to Find Segment Lengths In JKL, ZW = 7, and LX = 8.1. Find each length. 1a. KW 1b. LZ Problem-Solving Application The diagram shows the plan for a triangular piece of a mobile. 2. Find the average of the x-coordinates and the average of the y-coordinates of the vertices of PQR. Make a conjecture about the centroid of a triangle. Altitude of a triangle Orthocenter of the triangle Finding the Orthocenter 3. Show that the altitude of JK passes through the orthocenter of JKL. Q: What do you do when it rains? A: Coincide!
6 5-4 The Triangle Midsegment Theorem Midsegment Examining Midsegments in the Coordinate Plane 1. The vertices of RST are R 7,0, S 3,6, and T 9,2. M is the midpoint of RT, and N is the midpoint of ST. Show that MN RS and MN = 1 2 RS. Triangle Midsegment Theorem Using the Triangle Midsegment Theorem Find each measure. 2a. JL 2b. PM 2c. m MLK Indirect Measurement Application (Page 323) Anna wants to find the distance across the base of Capulin Volcano, and extinct volcano in New Mexico. She measures a triangle at one side of the volcano as shown in the diagram. What is AE? Suppose Anna s result in Example 3 is correct. To check it, she measures a second triangle. How many meters will she measure between H and F? Q: What id the visitor from Planet Metric demand? A: Take me to your liter!
7 Indirect proof 5-5 Indirect Proof and Inequalities in One Triangle 1. Writing an Indirect Proof Writing an Indirect Proof 1. Write an indirect proof that a triangle cannot have two right angles. Angle-Side Relationships in Triangles Theorem Hypothesis Conclusion Ordering Triangle Side Lengths and Angle Measures 2a. Write the angles in order from smallest to largest. 2b. Write the sides in order from shortest to longest.
8 Triangle Inequality Theorem Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 3a. 8, 13, 21 3b. 6.2, 7, 9 3c. t 2, 4t, t when t = 4 Finding Sides Lengths 4. The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side. Travel Application 5. The distance from San Marcos to Johnson City is 50 miles and the distance from Sequin to San Marcos is 22 miles. What is the range of distances from Sequin to Johnson City? Q: What do you call a prisoner s poem? A: A converse.
9 5-6 Inequalities in Two Triangles Inequalities in Two Triangles Theorem Hypothesis Conclusion Using the Hinge Theorem and Its Converse Compare the given measures. 1a. m EGH and m EGF 1b. BC and AB Entertainment Application The angle of the swings in a circular swing ride changes with the speed of the ride. The diagram shows the position of one swing at two different speeds. When the swing ride is at full speed, the chairs are furthest from the base of the swing tower. What can you conclude about the angles of the swings at full speed versus low speed? Explain.
10 Proving Triangle Relationships Write a two-column proof. 3a. Given: C is the midpoint of BD, m 1 = m 2, m 3 > m 4 Prove: AB > ED 3b. Given: SRT STR, TU > RU Prove: m TSU > m RSU Mr. Graves: Today we will study the Hinge Theorem Student: Do we need tools?
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