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Chapter 5 11 Glencoe Geometry

5-2 Practice Medians and Altitudes of Triangles Medians A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle intersect at the centroid of the triangle. The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. Example: In ᇞABC, U is the centroid and

BU = 16. Find UK and BK.

BU = ଶଷBK

16 = ଶଷBK

24 = BK

BU + UK = BK

16 + UK = 24

UK = 8

Exercises: In ᇞABC, AU = 16, BU = 12, and CF = 18. Find each measure. 1. UD 2. EU 3. CU 4. AD 5. UF 6. BE In ᇞCDE, U is the centroid, UK = 12, EM = 21, and UD = 9. Find each measure. 7. CU 8. MU 9. CK 10. JU 11. EU 12. JD

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Chapter 5 12 Glencoe Geometry

5-2 Practice (cont.)

Medians and Altitudes of Triangles Altitudes An altitude of a triangle is a segment from a vertex to the line containing the opposite side meeting at a right angle. Every triangle has three altitudes which meet at a point called the orthocenter. Example: The vertices of ᇞABC are A(1, 3), B(7, 7) and C(9, 3). Find the coordinates of the orthocenter of ᇞABC. Find the point where two of the three altitudes intersect. Find the equation of the altitude from A to ܥܤതതതത .

If ܥܤതതതത KDV�D�VORSH�RI�í���WKHQ�WKH�DOWLWXGH has a slope of ଵ

ଶ.

y – ݕଵ = m(x – ݔଵ) Point-slope form

y – 3 = ଵଶ(x – 1) m = ଵ

ଶ A(1, 3) = (ଵݕ ,ଵݔ) ,

y – 3 = ଵଶx – ଵ

ଶ Distributive Property

y = ଵଶx + ହ

ଶ Simplify.

Find the equation of the altitude from C to ܤܣ തതതതത.

If ܤܣതതതത has a slope of ଶଷ, then the altitude has a slope

of – ଷଶ.

y – ݕଵ = m(x – ݔଵ) Point-slope form

y – 3 = – ଷଶ(x – 9) m = – ଷ

ଶ C(9, 3) = (ଵݕ ,ଵݔ) ,

y – 3 = – ଷଶx + ଶ

ଶ Distributive Property

y = – ଷଶx + ଷଷ

ଶ Simplify.

Solve the system of equations and find where the altitudes meet.

y = ଵଶx + ହ

ଶ y = – ଷ

ଶx + ଷଷ

ଶ Original equations

ଵଶx + ହ

ଶ = ଷ

ଶx + ଷଷ

ଶ Substitute ଵ

ଶx + ହ

ଶ for y.

ହଶ �í�x + ଷଷ

ଶ Subtract ଵ

ଶx from each side.

í��� �í�x Subtract ଷଷଶ

from each side.

7 = x Divide each side by –2. y = ଵ

ଶx + ହ

ଶ = ଵ

ଶ (7) + ହ

ଶ =

ଶ + ହ

ଶ = 6

The coordinates of the orthocenter of ᇞABC are (7, 6). Exercises: COORDINATE GEOMETRY Find the coordinates of the orthocenter of the triangle with the given vertices. 1. J(1, 0), H(6, 0), I(3, 6) 2. S(1, 0), T(4, 7), U����í��