Ready to Go On? Skills Intervention 5-1 Perpendicular and Angle Bisectors

Transcription

1 Ready to Go On? Skills Intervention 5-1 Perpendicular and Angle isectors Find these vocabulary words in Lesson 5-1 and the Multilingual Glossary. equidistant focus Applying the Perpendicular isector Theorem and Its onverse Find each measure. A. DF Since D E, and DE, F is the 5.8 D F of DE by the onverse of the Theorem. Therefore, DF FE because of the definition of a 5.8 E. Substitute for FE. DF. VU TU VU because of the Theorem. Substitute the given measures for TU and VU and solve for x. T W V 1x + 14x 5 U 1x 1x 1x 5 14x 5 5 1x 8 x x Substitute the value of x to find VU. 14x 5 14( ) 5 Applying the Angle isector Theorem Find QR. Q R QR RS because of the Substitute 46 for RS. QR Theorem. P S 57 Holt Geometry

2 Ready to Go On? Skills Intervention 5- isectors of Triangles Find these vocabulary words in Lesson 5- and the Multilingual Glossary. concurrent point of concurrency circumcenter of a triangle circumscribed incenter of a triangle inscribed Using Properties of Perpendicular isectors P, PD and P are the perpendicular bisectors of LMN. A. Find PM. P is the of LMN. y the Theorem, P is 6. P L from the vertices of LMN, so PL PM. M 4.8 D N Substitute 7.7 for PL. PM. Find DN. y the definition of a, DM DN. Substitute 4.8 for DM. DN Finding the ircumcenter of a Triangle Find the circumcenter of D with vertices (8, 0), (0, 6), and D(0, 0). Graph the triangle. The equation for the perpendicular bisector of D is x. The equation for the perpendicular bisector of D is y. Find the intersection of the two equations. This point is the circumcenter of D: (, ). D O y x Using Properties of Angle isectors WT and WS are angle bisectors of STU. Find mtsv. T Since WS is the bisector of TSV, m (mtsw ). Substitute 18 for mtsw. mtsv ( ) S 18 W 7 58 V U 58 Holt Geometry

3 Find these vocabulary words in Lesson 5- and the Multilingual Glossary. Ready to Go On? Skills Intervention 5- Medians and Altitudes of Triangles median of a triangle centroid of a triangle altitude of a triangle orthocenter of a triangle Using the entroid to Find Segment Lengths In QRS, RP 1 and S 16. Find each length. A. SP Q P is called the of the triangle because it is the point of intersection of the of the triangle. The entroid Theorem states that the centroid of a triangle A P is located of the distance from each vertex to the of the opposite side. S R SP entroid Theorem SP ( ) Substitute the known value for S. SP. RA Simplify. RP entroid Theorem (1) ( ) Substitute known values for RP. RA To solve the equation, multiply both sides by.. AP RA Solve for RA. PA RA PA Substitute known values and solve. 59 Holt Geometry

4 Ready to Go On? Problem Solving Intervention 5- Medians and Altitudes of Triangles A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Sandra cuts a triangle with vertices W(0, 4), X(5, 0), and Y(10, ) from grid paper. At what coordinates should she place the tip of a pencil to balance the triangle? Understand the Problem 1. The of a triangle is called the center of gravity because it is the point where a triangular region will.. The answer to this problem will be the coordinates of the. Make a Plan. First, graph the triangle. 4. The centroid of the triangle is the point of intersection of the of the triangle. 5. Find the equations for two of the and find W 4 O y Y x the point of. Solve 6. Use the Midpoint Formula, M x 1 x, y 1 y, to find the midpoint, M, of WY. M 0, 4, 7. Use the Midpoint Formula to find the midpoint, N, of WX. N 0, 4 8. XM is a line. Its equation is x., 9. YN is a line. Its equation is y. 10. The coordinates of the centroid are. Look ack 11. Find the midpoint of XY. 1. Draw the line containing W and the midpoint. Does the point (5, ) lie on the line? 60 Holt Geometry

5 Ready To Go On? Skills Intervention 5-4 The Triangle Midsegment Theorem Find this vocabulary word in Lesson 5-4 and the Multilingual Glossary. midsegment of a triangle Using the Triangle Midsegment Theorem Find each measure. O A. Find., is a joins the of MNO because it is a segment that of two sides of the triangle. According to the Triangle Midsegment Theorem, 1 ( ). Since AM 15, OA, and OM. 15 A M 74 N Substitute 0 for OM. 1 ( ) Simplify to find.. Find ma. According to the Triangle Midsegment Theorem,. What kind of angles are OA and A? y the Alternate Interior Angles Theorem, you know that moa m. Substitute 74 for moa to find ma. ma. Find JL. XY is a joins the of JKL because it is a segment that of two sides of the triangle. According to the Triangle Midsegment Theorem, XY 1 ( ). 1 JL Substitute 5 for XY. (5) 1 JL Multiply both sides of the equation by to find JL. JL Simplify to find JL. J L 54 X 5 40 Y K 61 Holt Geometry

6 Ready to Go On? Problem Solving Intervention 5-4 The Triangle Midsegment Theorem The Triangle Midsegment Theorem can be used to determine measures that are difficult to determine. Karol wants to find the distance across the pond. He records the measurements in the diagram shown. Find the distance across the pond. Understand the Problem 1. You can use triangle to make indirect measurements of distances.. A midsegment of a triangle is a that joins the of two sides of the triangle.. The Triangle Midsegment Theorem states that a midsegment of a triangle is to a side of the triangle, and that its length is J 105 yd M 158 yd 105 yd the length of that side. 91 yd N K 158 yd L Make a Plan 4. Use the given information to determine whether MN is a midsegment of. 5. If MN is a midsegment, apply the Theorem to find, the distance across the pond. Solve 6. Since JM MK 105, M is the of. 7. Since JN NL 158, N is the of. 8. MN is the of JKL, and by the Triangle Midsegment Theorem, MN 1 ( ). 9. Substitute 91 for MN. 1 KL 10. Simplify to find KL. KL 11. What is the distance across the pond? Look ack 1. Work backward to check your answer. Divide KL by. KL Does this value equal MN? 6 Holt Geometry