) = (0, -2) Midpoint of AC ) = ( 2, MODULE. STUDY GUIDE REVIEW Special Segments in Triangles

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1 Houghton Mifflin Harcourt Publishing ompan STUDY GUIDE REVIEW Special Segments in Triangles Essential Question: How can ou use special segments in triangles to solve real-world problems? KEY EXMPLE (Lesson 8.1) Find the coordinates of the circumcenter of the triangle. oordinates: (-, -), (, 3), (, -) (-) ) = (, -) Midpoint of M ( = - +, is horizontal, so the line perpendicular to it is vertical and passes through the midpoint. The equation is =. M = + (, 3 + (-) ) = (, 1 is vertical, so the line perpendicular to it is horizontal and passes through the midpoint. The equation is = Find the equation of the line perpendicular to. ) Midpoint of 1. The coordinates of the circumcenter are (, 1 ). KEY EXMPLE (Lesson 8.) P and P are angle bisectors of, where P is the incenter of the triangle. The measure of is 56. The measure of is. Find the measures of P and P. Find the equation of the line perpendicular to. Since P is an angle bisector of, the measures of P and P are equal. Since the measure of is 56, the measure of P is 8. Since P is an angle bisector of, the measures of P and P are equal. Since the measure of is, the measure of P is 1. MODULE 8 Ke Vocabular altitude of a triangle (altura de un triángulo) centroid of a triangle (centroide de un triángulo) circumcenter of a triangle (circuncentro de un triángulo) circumscribed circle (círculo circunscrito) concurrent (concurrente) distance from a point to a line (distancia desde un punto hasta una línea) equidistant (equidistante) incenter of a triangle (incentro de un triángulo) inscribed circle (círculo inscrito) median of a triangle (mediana de un triángulo) midsegment of a triangle (segment medio de un triángulo) orthocenter of a triangle (ortocentro de un triángulo) point of concurrenc (punto de concurrencia ) Module 8 5 Stud Guide Review

2 KEY EXMPLE (Lesson 8.3) Find the coordinates of the centroid of the triangle. 6 oordinates: (-1, ), (3, 6), (, ) entroid: M = (, + 6 ) = (1, ) Midpoint of 6 m M = = - 3 Slope of line passing through midpoint and - = - ( - 1) 3 = Find the equation of the median from to M = (, + ) = ( 3, ) Midpoint of m M = 6 - = = 8 ( - 3) 3 = = = = 8 3 () - = 1 3 The coordinates of the centroid are (, 1 3 ). Slope of line passing through midpoint and Find the equation of the median. Set the equations equal to each other to find the intersection. KEY EXMPLE (Lesson 8.) DE is a midsegment of, and it is parallel to. If the length of D is 5 and the length of E is 3, find the lengths of D and E. DE is a midsegment of, so D is half of. D is the other half of. So, D = D = 5. DE is a midsegment of, so E is half of. E is the other half of. So, E = E = 3. Houghton Mifflin Harcourt Publishing ompan Module 8 6 Stud Guide Review

3 EXERISES Find the coordinates of the points. (Lesson 8.1) ircumcenter P, P, and P are angle bisectors of, where P is the incenter of the triangle. The measure of is. The measure of is 91. Find the measures of the angles. (Lesson 8.). P 3. P. P Find the coordinates of the points. (Lesson 8.3) Houghton Mifflin Harcourt Publishing ompan 5. entroid 6. Orthocenter DE, DF, and EF are midsegments of. Find the lengths of the segments. (Lesson 8.) 7. D E F D 7 F E Module 8 7 Stud Guide Review

4 MODULE PERFORMNE TSK What s the enter of the Triangle? The Teas Triangle Park in ran, Teas bills itself as being at the center of the Teas Triangle region. That is the region with the cities of Dallas, Houston, and San ntonio at the vertices of the triangle. The diagram shows a simple representation of the region with San ntonio located at the origin. The point also gives ou coordinates for the location of ran. So just how close is ran to the center of this triangle? efore ou tackle this problem, decide what ou think is the best measure of the triangle s center in this contet the centroid, circumcenter, or orthocenter? e prepared to support our decision. Start b listing in the space below the information ou will need to solve the problem. Then use our own paper to complete the task. e sure to write down all our data and assumptions. Then use graphs, numbers, words, or algebra to eplain how ou reached our conclusion D (13, 19) (135, 65) S (, ) H (19, ) Houghton Mifflin Harcourt Publishing ompan Module 8 8 Stud Guide Review

5 Read to Go On? Special Segments in Triangles Segments DE, EF, and DF are midsegments of Find the lengths of the indicated segments. (Lesson 8.1) 1.. F 3. DE. E EF Locate centroids, circumcenters, and incenters. (Lesson 8.) 7. Find the points of concurrenc of. a. Determine the coordinates of the centroid of. b. Determine the coordinates of the circumcenter of. D F 6 5 E Online Homework Hints and Help Etra Practice 8 Houghton Mifflin Harcourt Publishing ompan c. In what quadrant or on what ais does the incenter of lie? ESSENTIL QUESTION 8. Describe a triangle for which the centroid, circumcenter, incenter, and orthocenter are the same point. What features of this triangle cause these points to be concurrent and wh? Module 8 9 Stud Guide Review

6 MODULE 8 MIXED REVIEW ssessment Readiness 1. Given and altitude H, decide whether each statement is necessaril true about H. Select Yes or No for... H < H Yes No H < Yes No. H H Yes No H. YZ is the image of YX after a reflection across line M. hoose True or False for each statement. X Y W M Z. M is the angle bisector of XYZ. Yes No. XYZ is acute. Yes No. M is horizontal. Yes No 3. Given is equilateral, what can be determined about its centroid and circumcenter?. DE, EF, and DF are the midsegments of. How does the perimeter of DEF compare to the perimeter of? Eplain. D F E Houghton Mifflin Harcourt Publishing ompan Module 8 1 Stud Guide Review

KEY EXAMPLE (Lesson 23.1) Find the coordinates of the circumcenter of the triangle. Coordinates: A (-2, -2), B (2, 3), C (2, -2) 2) Midpoint of BC

KEY EXAMPLE (Lesson 23.1) Find the coordinates of the circumcenter of the triangle. Coordinates: A (-2, -2), B (2, 3), C (2, -2) 2) Midpoint of BC Houghton Mifflin Harcourt Publishing ompan STUDY GUIDE REVIEW Special Segments in Triangles Essential Question: How can ou use special segments in triangles to solve real-world problems? KEY EXMPLE (Lesson

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