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Vocabulary MedianA segment with endpoints being a vertex of a triangle and the midpoint of the opposite side. AltitudeA segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. An altitude may be out side the circle.

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Concept (also called center of mass or center of gravity) From Latin: centrum - "center", and Greek: -oid -"like"

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

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Example 1 Use the Centroid Theorem Answer: YP = 8; PV = 4 YP + PV= YVSegment Addition 8 + PV= 12YP = 8 PV= 4Subtract 8 from each side.

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

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

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Example 2 Use the Centroid Theorem Centroid Theorem Segment Addition and Substitution CG = 4 Distributive Property

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Example 2 Use the Centroid Theorem Answer: GE = 2 Subtract GE from each side. __ 1 3

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

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Example 3 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 artists 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? UnderstandYou need to find the centroid of the triangle. This is the point at which the triangle will balance.

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Example 3 Find the Centroid on a Coordinate Plane SolveGraph ABC. PlanGraph and label the triangle with vertices (1, 4), (3, 0), and (3, 8). 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.

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Example 3 Find the Centroid on a Coordinate Plane Find the midpoint D of side BC. Graph point D.

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

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Example 3 Find the Centroid on a Coordinate Plane The centroid is the distance. So, the centroid is (2) or units to the right of A. The coordinates are.

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Example 3 Find the Centroid on a Coordinate Plane Answer: The artist should place the pole at the point CheckCheck the distance of the centroid from point D (3, 4). The centroid should be (2) or units to the left of D. So, the coordinates of the centroid is. __ 1 3 2 3

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A.A B.B C.C D.D 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.(, 2) B.(, 2) C.(1, 2) D.(0, 4) __ 7 3 5 3

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Concept From Greek: orthos - "straight, true, correct, regular

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

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Example 4 Find the Orthocenter on a Coordinate Plane Find an equation of the altitude from The slope of so the slope of an altitude is Point-slope form Distributive Property Add 1 to each side.

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Example 4 Find the Orthocenter on a Coordinate Plane Point-slope form Distributive Property Subtract 3 from each side. Next, find an equation of the altitude from I to The slope of so the slope of an altitude is 6.

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Example 4 Find the Orthocenter on a Coordinate Plane Equation of altitude from J Multiply each side by 5. Add 105 to each side. Add 4x to each side. Divide each side by 26. Substitution, Then, solve a system of equations to find the point of intersection of the altitudes.

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Example 4 Find the Orthocenter on a Coordinate Plane Replace x with in one of the equations to find the y-coordinate. Multiply and simplify. Rename as improper fractions.

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Example 4 Find the Orthocenter on a Coordinate Plane Answer: The coordinates of the orthocenter of HIJ are

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

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Concept