Identify and use medians in triangles. You identified and used perpendicular and angle bisectors in triangles. Identify and use medians in triangles. Identify and use altitudes in triangles. Then/Now

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A. Find ST if S is the incenter of ΔMNP. Use the Incenter Theorem A. Find ST if S is the incenter of ΔMNP. Example 4

A. Find BD if D is the circumcenter of ΔABC. Use the Circumcenter Theorem A. Find BD if D is the circumcenter of ΔABC. Example 4

B. Find mSPU if S is the incenter of ΔMNP. Use the Incenter Theorem B. Find mSPU if S is the incenter of ΔMNP. Example 4

A. Find the measure of GF if D is the incenter of ΔACF. B. 144 C. 8 D. 65 Example 4

Example 4End of the Lesson B. Find the measure of BCD if D is the incenter of ΔACF. A. 58° B. 116° C. 52° D. 26° Example 4End of the Lesson

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In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Example 1

In ΔLNP, R is the centroid and LO = 30. Find LR and RO. A. LR = 15; RO = 15 B. LR = 20; RO = 10 C. LR = 17; RO = 13 D. LR = 18; RO = 12 Example 1

Use the Centroid Theorem In ΔABC, CG = 4. Find GE. Example 2

In ΔJLN, JP = 16. Find PM. A. 4 B. 6 C. 16 D. 8 Example 2

Find the Centroid on a Coordinate Plane SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist’s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance? Understand You need to find the centroid of the triangle. This is the point at which the triangle will balance. Example 3

Find the Centroid on a Coordinate Plane Plan STEP 1: Graph and label the triangle with vertices at A(1, 4), B(3, 0), and C(3, 8). Example 3

Find the Centroid on a Coordinate Plane Use the Midpoint Theorem to find the midpoint of one of the sides of the triangle. The centroid is two-thirds the distance from the opposite vertex to that midpoint. STEP 2: Find the midpoint D of BC. And Graph point D. Example 3

STEP 3: Find the distance. Find the Centroid on a Coordinate Plane STEP 3: Find the distance. Notice that is a horizontal line. The distance from D(3, 4) to A(1, 4) is 3 – 1 or 2 units. Example 3

STEP 4: Solve Soooo, how did they get 7/3? Find the Centroid on a Coordinate Plane The centroid P is the distance. So, the centroid is (2) or units to the right of A. The coordinates are . STEP 4: Solve Soooo, how did they get 7/3? P Example 3

BASEBALL A fan of a local baseball team is designing a triangular sign for the upcoming game. In his design on the coordinate plane, the vertices are located at (–3, 2), (–1, –2), and (–1, 6). What are the coordinates of the point where the fan should place the pole under the triangle so that it will balance? A. B. C. (–1, 2) D. (0, 4) Example 3

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Find the Orthocenter on a Coordinate Plane COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of ΔHIJ. Example 4

Find an equation of the altitude from The slope of Find the Orthocenter on a Coordinate Plane Find an equation of the altitude from The slope of so the slope of an altitude is Example 4

Next, find an equation of the altitude from I to The Find the Orthocenter on a Coordinate Plane Next, find an equation of the altitude from I to The slope of so the slope of an altitude is –6. Example 4

Find the Orthocenter on a Coordinate Plane Then, solve a system of equations to find the point of intersection of the altitudes. Example 4

COORDINATE GEOMETRY The vertices of ΔABC are A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of ΔABC. A. (1, 0) B. (0, 1) C. (–1, 1) D. (0, 0) Example 4

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