23.1 Perpendicular Bisectors of Triangles

Transcription

1 Name lass Date 3.1 Perpendicular isectors of Triangles Essential Question: How can ou use perpendicular bisectors to find the point that is equidistant from all the vertices of a triangle? Resource Locker Eplore onstructing a ircumscribed ircle circle that contains all the vertices of a polgon is circumscribed about the polgon. In the figure, circle is circumscribed about XYZ, and circle is called the circumcircle of XYZ. The center of the circumcircle is called the circumcenter of the triangle. In the following activit, ou will construct the circumcircle of PQR. op the triangle onto a separate piece of paper. X Y Z The circumcircle will pass through P, Q, and R. So, the center of the circle must be equidistant from all three points. In particular, the center must be equidistant from Q and R. Q The set of points that are equidistant from Q and R is called the of QR. Use a compass and straightedge to construct the set of points. R The center must also be equidistant from P and R. The set of points that are equidistant from P and R is called the of PR. Use a compass and straightedge to construct the set of points. P Houghton Mifflin Harcourt Publishing ompan The center must lie at the intersection of the two sets of points ou constructed. Label the point. Then place the point of our compass at and open it to distance P. Draw the circumcircle. Module Lesson 1

2 Reflect 1. Make a Prediction Suppose ou started b constructing the set of points equidistant from P and Q and then constructed the set of points equidistant from Q and R. Would ou have found the same center? heck b doing this construction.. an ou locate the circumcenter of a triangle without using a compass and straightedge? Eplain. Eplain 1 Proving the oncurrenc of a Triangle s Perpendicular isectors Three or more lines are concurrent if the intersect at the same point. The point of intersection is called the point of concurrenc. You saw in the Eplore that the three perpendicular bisectors of a triangle are concurrent. Now ou will prove that the point of concurrenc is the circumcenter of the triangle. That is, the point of concurrenc is equidistant from the vertices of the triangle. ircumcenter Theorem The perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the vertices of the triangle. P = P = P P Eample 1 Prove the ircumcenter Theorem. Given: Lines l, m, and n are the perpendicular bisectors of,, and, respectivel. P is the intersection of l, m, and n. Prove: P = P = P P is the intersection of l, m, and n. Since P lies on the of, P = P b the Theorem. Similarl, P lies on the of, so = P. Therefore, P = = b the Propert of Equalit. l m P n Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

3 Reflect 3. Discussion How might ou determine whether the circumcenter of a triangle is alwas inside the triangle? Make a plan and then determine whether the circumcenter is alwas inside the triangle. Eplain Using Properties of Perpendicular isectors You can use the ircumcenter Theorem to find segment lengths in a triangle. Eample KZ, LZ, and MZ are the perpendicular bisectors of GHJ. Use the given information to find the length of each segment. Note that the figure is not drawn to scale. H K Z L G M J Given: ZM = 7, ZJ = 5, HK = 0 Find: ZH and HG Z is the circumcenter of GHJ, so ZG = ZH = ZJ. Houghton Mifflin Harcourt Publishing ompan ZJ = 5, so ZH = 5. K is the midpoint of GH, so HG = KH = 0 = 0. Given: ZH = 85, MZ = 13, HG = 136 Find: KG and ZJ K is the of HG, so KG = HG = =. Z is the of GHJ, so ZG = =. ZH =, so ZJ =. Module Lesson 1

4 Reflect. In, is a right angle and D is the circumcenter of the triangle. If D = 6.5, what is? Eplain our reasoning. D Your Turn KZ, LZ, and MZ are the perpendicular bisectors of GHJ. op the sketch and label the given information. Use that information to find the length of each segment. Note that the figure is not drawn to scale. H K Z L G M J 5. Given: ZG = 65, HL = 63, ZL = 16 Find: HJ and ZJ 6. Given: ZM = 5, ZH = 65, GJ = 10 Find: GM and ZG Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

5 Eplain 3 Finding a ircumcenter on a oordinate Plane Given the vertices of a triangle, ou can graph the triangle and use the graph to find the circumcenter of the triangle. Eample 3 Graph the triangle with the given vertices and find the circumcenter of the triangle. R ( -6, 0 ), S ( 0, ), O ( 0, 0 ) Step 1: Graph the triangle. = -3 6 S Step : Find equations for two perpendicular bisectors. Side RO is on the -ais, so its perpendicular bisector is vertical: the line = -3. Side SO is on the -ais, so its perpendicular bisector = (-3, ) R O is horizontal: the line =. Step 3: Find the intersection of the perpendicular bisectors. The lines = -3 and = intersect at (-3, ). (-3, ) is the circumcenter of ROS. (-1, 5), (5, 5), (5, -1) Houghton Mifflin Harcourt Publishing ompan Step 1 Graph the triangle. Step Find equations for two perpendicular bisectors. Side is, so its perpendicular bisector is vertical. The perpendicular bisector of is the line. Side is, so the perpendicular bisector of is the horizontal line. Step 3 Find the intersection of the perpendicular bisectors. The lines and intersect at. is the circumcenter of Module Lesson 1

6 Reflect 7. Draw onclusions ould a verte of a triangle also be its circumcenter? If so, provide an eample. If not, eplain wh not. Your Turn Graph the triangle with the given vertices and find the circumcenter of the triangle. 8. Q (-, 0), R (0, 0), S (0, 6) 9. K (1, 1), L (1, 7), M (6, 1) Elaborate 10. compan that makes and sells biccles has its largest stores in three cities. The compan wants to build a new factor that is equidistant from each of the stores. Given a map, how could ou identif the location for the new factor? 11. sculptor builds a mobile in which a triangle rotates around its circumcenter. Each verte traces the shape of a circle as it rotates. What circle does it trace? Eplain. Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

7 1. What If? Suppose ou are given the vertices of a triangle PQR. You plot the points in a coordinate plane and notice that PQ is horizontal but neither of the other sides is vertical. How can ou identif the circumcenter of the triangle? Justif our reasoning. 13. Essential Question heck-in How is the point that is equidistant from the three vertices of a triangle related to the circumcircle of the triangle? Evaluate: Homework and Practice onstruct the circumcircle of each triangle. Label the circumcenter P. 1.. Online Homework Hints and Help Etra Practice Houghton Mifflin Harcourt Publishing ompan 3.. Module Lesson 1

8 omplete the proof of the ircumcenter Theorem. Use the diagram for Eercise 5 8. ZD, ZE, and ZF are the perpendicular bisectors of. Use the given information to find the length of each segment. Note that the figure is not drawn to scale. 5. Given: DZ = 0, Z = 85, F = 77 D Z F Find: Z and E 6. Given: FZ = 36, Z = 85, = 150 Find: D and Z 7. Given: Z = 85, ZE = 51 Find: (Hint: Use the Pthagorean Theorem.) 8. nalze Relationships How can ou write an algebraic epression for the radius of the circumcircle of in Eercises 6 8? Eplain. Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

9 omplete the proof of the ircumcenter Theorem. 9. Given: Lines l, m, and n are the perpendicular bisectors of,, and, respectivel. P is the intersection of l, m, and n. Prove: P = P = P l P n m Statements Reasons 1. Lines l, m, and n are the perpendicular bisectors of,, and. 1.. P is the intersection of l, m, and n.. 3. P = 3. P lies on the perpendicular bisector of.. = P. P lies on the perpendicular bisector of. 5. P = = PK, PL, and PM are the perpendicular bisectors of sides,, and. Tell whether the given statement is justified b the figure. Select the correct answer for each lettered part. a. K = K Justified Not Justified b. P = P Justified Not Justified c. PM = PL Justified Not Justified d. L = 1 Justified Not Justified e. PK = KD Justified Not Justified P K M D L Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

10 Graph the triangle with the given vertices and find the circumcenter of the triangle. 11. D (-5, 0), E (0, 0), F (0, 7) Q (3, ), R (7, ), S (3, -) Represent Real-World Problems For the net Fourth of Jul, the towns of shton, radford, and learview will launch a fireworks displa from a boat in the lake. Draw a sketch to show where the boat should be positioned so that it is the same distance from all three towns. Justif our sketch. learview shton radford H.O.T. Focus on Higher Order Thinking 1. nalze Relationships Eplain how can ou draw a triangle JKL whose circumcircle has a radius of 8 centimeters. Houghton Mifflin Harcourt Publishing ompan Module Lesson 1

11 15. Persevere in Problem Solving ZD, ZE and ZF are the perpendicular bisectors of, which is not drawn to scale. D Z F E a. Suppose that Z = 15, ZD = 100, and ZF = 17. How can ou find and? b. Find and. c. an ou find? If so, eplain how and find. If not, eplain wh not. Houghton Mifflin Harcourt Publishing ompan 16. Multiple Representations Given the vertices (-, -), (, 0), and (, ) of a triangle, the graph shows how ou can use a graph and construction to locate the circumcenter P of the triangle. You can draw the perpendicular bisector of and construct the perpendicular bisector of. onsider how ou could identif P algebraicall. a. The perpendicular bisector of passes through its midpoint. Use the Midpoint Formula to find the midpoint of. b. What is the slope m of the perpendicular bisector of? Eplain how ou found it (-, -) - (, ) (, 0) 6 c. Write an equation of the perpendicular bisector of and eplain how ou can use it find P. Module Lesson 1

12 Lesson Performance Task landscape architect wants to plant a circle of flowers around a triangular garden. She has sketched the triangle on a coordinate grid with vertices at (0, 0), (8, 1), and (18, 0) (8, 1) (0, 0) (18, 0) Eplain how the architect can find the center of the circle that will circumscribe triangle. Then find the radius of the circumscribed circle. Houghton Mifflin Harcourt Publishing ompan Module Lesson 1