[PDF] 5-2 Study Guide and Intervention - Medians and Altitudes of Triangles





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5-2 Study Guide and Intervention - Medians and Altitudes of Triangles

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

Chapter 5 11 Glencoe Geometry

5-2 Study Guide and Intervention

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.

7BK 7BK

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

8 6 12 24 6 18 24 7

36 4.5

14 13.5

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

Chapter 5 12 Glencoe Geometry

5-2 Study Guide and Intervention (continued)

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. has a slope of ଵ 6. y ݕଵ = m(x ݔଵ) Point-slope form y 3 = ଵ

6(x 1) m = ଵ

6, (ݔଵ, ݕଵ) = A(1, 3)

y 3 = ଵ

6x ଵ

6 Distributive Property

y = ଵ

6x + ହ

6 Simplify.

Find the equation of the altitude from

7, then the altitude has a slope

of ଷ 6. y ݕଵ = m(x ݔଵ) Point-slope form y 3 = ଷ

6(x 9) m = ଷ

6, (ݔଵ, ݕଵ) = C(9, 3)

y 3 = ଷ

6 Distributive Property

y = ଷ

6x + ଷଷ

6 Simplify.

Solve the system of equations and find where the altitudes meet. y = ଵ

6x + ହ

6 y = ଷ

6x + ଷଷ

6 Original equations

6x + ହ

6 = ଷ

6x + ଷଷ

6 Substitute ଵ

6x + ହ

6 for y.

6 x + ଷଷ

6 Subtract ଵ

6x from each side.

x Subtract ଷଷ

6 from each side.

7 = x Divide each side by 2.

y = ଵ

6x + ହ

6 = ଵ

6 (7) + ହ

6 = ଻

6 + ହ

6 = 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

(3, 1) (1, 0)quotesdbs_dbs31.pdfusesText_37
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