Notes on Perp. Bisectors & Circumcenters - Page 1

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1 Notes on Perp. isectors & ircumcenters - Page 1 Name perpendicular bisector of a triangle is a line, ray, or segment that intersects a side of a triangle at a 90 angle and at its midpoint. onsider to the left. Suppose that point M is the midpoint of side. M If one is constructing the perpendicular bisector to side, then it must pass through the midpoint (M). However, it must also be perpendicular to side. For example, line p to the right is NOT a perpendicular bisector. line p ven though it passes thru the midpoint, it is clearly not perpendicular to side. See that the angle formed by the intersection is obviously not 90 in measure. M NOT a perpendicular bisector line r In contrast, line r to the left is a perpendicular bisector. Note how it passes thru the midpoint, N it intersects side at a 90 angle. M perpendicular bisector R T For the triangles to the left and right, the midpoint of side RT has been represented by the dot. S T S Sketch the perpendicular bisector to side RT in each example. In the examples of perpendicular bisectors above, note that the perpendicular bisector does not pass through the vertex of the triangle. Perpendicular bisectors for triangles rarely pass through a vertex of the triangle. R

2 For Questions 1-4, draw all three perpendicular bisectors for the given triangle. To assist you, the midpoint of each side has already been found. To draw each perpendicular bisector, simply draw a perpendicular line through each midpoint. o not force them to go through vertices! In the problems above, you should notice that the three perpendicular bisectors of a triangle intersect at exactly one point. Therefore, they are concurrent. circumcenter The circumcenter is the point of concurrency for the perpendicular bisectors of a triangle. The triangles in Questions 1 & 2 above are both acute triangles. In both cases, the circumcenter is located inside of the triangle. In the figure to the left, the perpendicular bisectors have been drawn as dashed lines. s you can see, they intersect at exactly one point called the circumcenter. Similarly, the triangle to the right is an acute triangle. Notice how all three perpendicular bisectors have been drawn, and that they intersect inside of the triangle. The circumcenter of an acute triangle is always located inside of the triangle.

3 Notes on Perp. isectors & ircumcenters - Page 2 Name The triangle in Question 3 on the previous page was a right triangle. Notice that the circumcenter is on the triangle. Similarly, the three perpendicular bisectors have been constructed for the right triangle (pictured to the left). gain, observe how the circumcenter is located on the triangle. In fact, in both cases, the circumcenter is at a specific location on the right triangle. The circumcenter of a right triangle is always located at the midpoint of the hypotenuse. For Questions 5-8, place a dot at the approximate location of the circumcenter without drawing the perpendicular bisectors. See Question 5 as an example midpoint of hypotenuse If you look at Question 4 on the previous page, it was an obtuse triangle. Its circumcenter was located outside of the triangle. The triangle pictured to the right is also obtuse. Its perpendicular bisectors have been drawn. gain, you can notice that the circumcenter is located outside of the triangle. The circumcenter for an obtuse triangle is always located outside of the triangle. The circumcenter is the same distance away from each of the vertices of a triangle. X Y For example, suppose point is the circumcenter of XYZ (right). ccording the property above, point is the same distance away from vertex X, as it is from vertex Y, as it is from vertex Z. Z

4 In the figure to the right, point is the circumcenter of Suppose J 2x 11, and K 6x 2. What is the value of x? JHK. J ccording to the previous property, point is the same distance from J as it is from K. Thus, J K. So, these two expressions will be equated, and we will solve for x: 2x 11 6x x x 3.25 line r Finally, since a perpendicular bisector creates right angles, it can form right triangles. This means the Pythagorean Theorem can occasionally be used. K H 6 N 8 For example, suppose line r is a perpendicular bisector to side to the left. If N 6, and N 8, what is? If one focuses on the bottom right portion of the figure above, then you can see the right triangle formed. It has been displayed to the right. If side is labeled c, then, using the Pythagorean Thm.: c N c 2 c 10 Hence, = 10 units. Homework on Perpendicular isectors & ircumcenters For Questions 1-3, give the location of the requested point based on the type of triangle. 1. acute triangle 2. right triangle 3. obtuse triangle circumcenter: circumcenter: circumcenter:

5 Homework on Perp. isectors & ircumcenters - ont. Name For Questions 7-8, consider the figure to the right. Triangle F is a right triangle, and line G is the perpendicular bisector of side F. 7. True or False: G = GF. G 8. True or False: G must be a right triangle. F O \ For Questions 15-16, consider the figure to the left. Let point S be the circumcenter of POR. lso, let SO 9x 1, and PS 11x 7. P S 15. What is the value of x? R 16. What is the length of SR? 17. True or False: The three perpendicular bisectors of a triangle are always concurrent. r In the figure to the left, line r is a perpendicular bisector of X side ZY. Let TW = 5, and WY = 12. Use the figure to answer Questions 18 and 19. T Y 18. What is WZ? Z W 19. What is ZT?

6 Notes on ngle isectors & Incenters Name onsider to the left. Its measure is 50. line m 50 Now, look to the right. Note how line m cuts into two smaller, but congruent angles. ssentially, the line cut in half, creating two angles with measures of 25. Line m is an angle bisector n angle bisector is a segment, ray or line that passes thru the vertex of an angle and divides the angle into two smaller, congruent angles. triangle can have angle bisectors. line n 72 For example, consider F (left) in which measures F To the right, line n serves as an angle bisector to. Notice how it cut the angle into two equal halves. F G onsider GJH to the left. K How do you know that ray HK does not bisect GHJ in the triangle? H J n angle bisector must divide an angle into two smaller angles of the same measure. Suppose line QM bisects M in the triangle to the right. If m LMQ 6x 4, and m NMQ 7x 5, then what is the value of x? Q N 7x - 5 Thus, what is the measure of LMQ? L 6x + 4 M

7 s you may have guessed, the lines containing the three angle bisectors of a triangle are concurrent. The point of concurrency for the angle bisectors of a triangle is known as the incenter. incenter In the figure to the left, the three angle bisectors for the triangle have been constructed. They intersect at exactly one point called the incenter. In the space below, the three angle bisectors have been drawn for an acute triangle, a right triangle, and an obtuse triangle. s you can see, the incenter is inside of all three triangles. cute Right Obtuse I I I The incenter is always located inside of the triangle. There is one important characteristic about all incenters: The incenter of a triangle is the same distance away from all three sides of the triangle. Please understand - this is different than the circumcenter (studied previously). The circumcenter is the same distance away from all three of the triangle. I For example, the incenter (I) has been located for the triangle to the left. To the right, you can see that the incenter is, in fact, the same distance from each side. same distance I same distance same distance

8 Homework on ngle isectors & Incenters Name 1. Point U is the incenter of triangle RST. If the distance from point U to side RS is 5 cm, then what is the distance from point U to side TR? If needed, draw a picture. F W 2. In the figure to the left, FW is an angle bisector of F. Name the angle that is congruent to FW. You must use three letters in the naming of the angle. 3. Fill in the blank: The circumcenter of a triangle is equidistant from the of the triangle. 4. What is the definition of concurrent lines?