Bisectors in Triangles

Transcription

1 5-3 isectors in riangles ontent tandard G..3 onstruct the inscribed and circumscribed circles of a triangle... Objective o identify properties of perpendicular bisectors and angle bisectors an you conjecture any other properties that the perpendicular lines might have? HIL I onstruct a circle and label its center. hoose any three points on the circle and connect them to form a triangle. raw three lines from such that each line is perpendicular to one side of the triangle. What conjecture can you make about the two segments into which each side of the triangle is divided? Justify your reasoning. In the olve It, the three lines you drew intersect at one point, the center of the circle. When three or more lines intersect at one point, they are concurrent. he point at which they intersect is the point of concurrency. Lesson Vocabulary concurrent c point of concurrency circumcenter of a triangle circumscribed about incenter of a triangle inscribed in ssential Understanding For any triangle, certain sets of lines are always concurrent. wo of these sets of lines are the perpendicular bisectors of the triangle s three sides and the bisectors of the triangle s three angles. heorem 5-6 oncurrency of erpendicular isectors heorem heorem he perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. iagram ymbols erpendicular bisectors,, and are concurrent at. he point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. ince the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. ou say the circle is circumscribed about the triangle. Lesson 5-3 isectors in riangles 301

2 roof roof of heorem 5-6 Given: rove: roof: Lines, m, and n are the perpendicular bisectors of the sides of. is the intersection of lines and m. Line n contains point, and. point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. oint is on, which is the perpendicular bisector of, so. Using the same reasoning, since is on m, and m is the perpendicular bisector of,. hus, by the ransitive roperty. ince, is equidistant from the endpoints of. hen, by the converse of the erpendicular isector heorem, is on line n, the perpendicular bisector of. m n he circumcenter of a triangle can be inside, on, or outside a triangle. cute triangle ight triangle Obtuse triangle roblem 1 Finding the ircumcenter of a riangle What are the coordinates of the circumcenter of the triangle with vertices (0, 6), O(0, 0), and (4, 0)? Find the intersection point of two of the triangle s perpendicular bisectors. Here, it is easiest to find the perpendicular bisectors of O and O. oes the location of the circumcenter make sense? es, O is a right triangle, so its circumcenter should lie on its hypotenuse. tep 1 (0, 3) is the midpoint of O. he line through (0, 3) that is perpendicular to O is y 3. tep 3 Find the point where the two perpendicular bisectors intersect. x 2 and y 3 intersect at (2, 3). y O 4 x 6 tep 2 (2, 0) is the midpoint of O. he line through (2, 0) that is perpendicular to O is x 2. he coordinates of the circumcenter of the triangle are (2, 3). Got It? 1. What are the coordinates of the circumcenter of the triangle with vertices (2, 7), (10, 7), and (10, 3)? 302 hapter 5 elationships Within riangles

3 roblem 2 Using a ircumcenter How do you find a point equidistant from three points? s long as the three points are noncollinear, they are vertices of a triangle. Find the circumcenter of the triangle. town planner wants to locate a new fire station equidistant from the elementary, middle, and high schools. Where should he locate the station? lementary chool own ark High chool iddle chool H he three schools form the vertices of a triangle. he planner should locate the fire station at, the point of concurrency of the perpendicular bisectors of H. his point is the circumcenter of H and is equidistant from the three schools at,, and H. Got It? 2. In roblem 2, the town planner wants to place a bench equidistant from the three trees in the park. Where should he place the bench? H heorem 5-7 oncurrency of ngle isectors heorem heorem he bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle. iagram ymbols ngle bisectors,, and are concurrent at. ou will prove heorem 5-7 in xercise 24. he point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. For any triangle, the incenter is always inside the triangle. In the diagram, points,, and are equidistant from, the incenter of. is the center of the circle that is inscribed in the triangle. Lesson 5-3 isectors in riangles 303

4 roblem 3 Identifying and Using the Incenter of a riangle lgebra G 2x 7 and GF x 4. What is G? What is the distance from a point to a line? he distance from a point to a line is the length of the perpendicular segment that joins the point to the line. G is the incenter of because it is the point of concurrency of the angle bisectors. y the oncurrency of ngle isectors heorem, the distances from the incenter to the three sides of the triangle are equal, so G GF G. Use this relationship to find x. 2x 7 x 4 G GF 2x x 11 dd 7 to each side. G F x 11 ubtract x from each side. Now find GF. GF x ubstitute 11 for x. ince GF G, G 15. Got It? 3. a. QN 5x 36 and Q 2x 51. What is QO? b. easoning Is it possible for Q to equal 50? xplain. O K Q N J L Lesson heck o you know HOW? 1. What are the coordinates of the circumcenter of the following triangle? 4 2 y O 2 2. In the figure at the right, V 3x 12 and U 5x 24. What is the value of x? 4 6 U V x o you UNN? 3. Vocabulary triangle s circumcenter is outside the triangle. What type of triangle is it? 4. easoning ou want to find the circumcenter of a triangle. Why do you only need to find the intersection of two of the triangle s perpendicular bisectors, instead of all three? 5. rror nalysis our friend sees the triangle at the right and concludes that. What is the error in your friend s reasoning? 6. ompare and ontrast How are the circumcenter and incenter of a triangle alike? How are they different? HIL I Q 304 hapter 5 elationships Within riangles

5 ractice and roblem-olving xercises HIL I ractice oordinate Geometry Find the coordinates of the circumcenter of each triangle. ee roblem y 8. y 2 O x 2 2 x 4 2 O 2 4 oordinate Geometry Find the coordinates of the circumcenter of. 9. (0, 0) (3, 0) (3, 2) 10. (0, 0) (4, 0) (4, 3) 11. ( 4, 5) ( 2, 5) ( 2, 2) 12. ( 1, 2) ( 5, 2) ( 1, 7) 13. (1, 4) (1, 2) (6, 2) 14. ity lanning opy the diagram of the beach. how where town officials should place a recycling barrel so that it is equidistant from the lifeguard chair, the snack bar, and the volleyball court. xplain. ee roblem 2. nack bar Volleyball court Lifeguard chair Name the point of concurrency of the angle bisectors. ee roblem Find the value of x (x 3) 6 and 5(2x 6). 4x 1 6x 5 L N Lesson 5-3 isectors in riangles 305

6 pply 19. hink bout a lan In the figure at the right, is the incenter of isosceles. What type of triangle is? xplain. What segments determine the incenter of a triangle? What do you know about the base angles of an isosceles triangle? onstructions raw a triangle that fits the given description. hen construct the inscribed circle and the circumscribed circle. escribe your method. 20. right triangle, F 21. obtuse triangle, U 22. lgebra In the diagram at the right, G is the incenter of F, m F 60, and m F 2 m F. What are m G, m GF, and m GF? G 23. Writing Ivars found an old piece of paper inside an antique book. It read, From the spot I buried Olaf s treasure, equal sets of paces did I measure; each of three directions in a line, there to plant a seedling Norway pine. I could not return for failing health; now the hounds of Haiti guard my wealth. Karl fter searching aribbean islands for five years, Ivars found an island with three tall Norway pines. How might Ivars find where Karl buried Olaf s treasure? F 24. Use the diagram at the right to prove the oncurrrency of ngle roof isectors heorem. F Given: ays, m, and n are bisectors of the angles of. is the intersection of rays and m,,, and F. rove: ay n contains point, and F. n m 25. Noise ontrol ou are trying to talk to a friend on the phone in a busy bus station. he buses are so loud that you can hardly hear. eferring to the figure at the right, should you stand at or to be as far as possible from all the buses? xplain. easoning etermine whether each statement is true or false. If the statement is false, give a counterexample. 26. he incenter of a triangle is equidistant from all three vertices. 27. he incenter of a triangle always lies inside the triangle. 28. ou can circumscribe a circle about any three points in a plane. 29. If point is the circumcenter of Q and the circumcenter of Q, then and must be the same point. 306 hapter 5 elationships Within riangles

7 hallenge 30. easoning xplain why the circumcenter of a right triangle is on one of the triangle s sides. etermine whether each statement is always, sometimes, or never true. xplain. 31. It is possible to find a point equidistant from three parallel lines in a plane. 32. he circles inscribed in and circumscribed about an isosceles triangle have the same center. tandardized est rep / 33. Which of the following statements is false? he bisectors of the angles of a triangle are concurrent. he midsegments of a triangle are concurrent. he perpendicular bisectors of the sides of a triangle are concurrent. Four lines intersecting in one point are concurrent. 34. What type of triangle is U? right isosceles obtuse scalene 42 acute isosceles acute scalene 35. Which statement is logically equivalent to the following statement? 45 U If a triangle is right isosceles, then it has exactly two acute angles. If a triangle is right isosceles, then it has one right angle. If a triangle has exactly two acute angles, then it is right isosceles. If a triangle does not have exactly two acute angles, then it is not right isosceles. K If a triangle is not right isosceles, then it does not have a right angle. V hort esponse 36. efer to the figure at the right above. xplain in two different ways why V is the angle bisector of KV. ixed eview Use the figure at the right for xercises 37 and Find the value of x. 2x 1 7 3x 5 ee Lesson Find the length of. Get eady! o prepare for Lesson 5-4, do xercises 39 and 40. Find the coordinates of the midpoint of with the given endpoints. 39. (3, 0), (3, 16) 40. (6, 8), (4, 1) ee Lesson 1-7. Lesson 5-3 isectors in riangles 307