Download 5-2 Practice (cont.)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________
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 △ABC, U is the centroid and
BU = 16. Find UK and BK.
2
3
BU = BK
2
3
16 = BK
24 = BK
BU + UK = BK
16 + UK = 24
UK = 8
Exercises:
In △ABC, AU = 16, BU = 12, and CF = 18. Find each measure.
1. UD
2. EU
3. CU
4. AD
5. UF
6. BE
In △CDE, 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
Chapter 5
11
Glencoe Geometry
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
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 △ABC are A(1, 3),
B(7, 7) and C(9, 3). Find the coordinates of the
orthocenter of △ABC.
Find the point where two of the three altitudes intersect.
���� .
Find the equation of the altitude from A to 𝐵𝐵
Find the equation of the altitude from
C to �����
𝐴𝐴 .
���� has a slope of −2, then the altitude
If 𝐵𝐵
1
has a slope of .
2
y – 𝑦1 = m(x – 𝑥1 )
3
2
of – .
y – 𝑦1 = m(x – 𝑥1 )
Point-slope form
1
y – 3 = (x – 1)
2
1
1
y–3= x–
2
2
1
5
y= x+
2
2
1
m = , (𝑥1 , 𝑦1 ) = A(1, 3)
y–
2
Distributive Property
y–
Simplify.
Solve the system of equations and find where the altitudes meet.
1
2
y= x+
5
2
3
2
y=– x+
1
x
2
33
2
5 3
33
2 2
2
5
33
= −2x +
2
2
+ = x+
−14 = −2x
7=x
1
2
5
2
1
2
5
2
2
3
���� has a slope of , then the altitude has a slope
If 𝐴𝐴
3
3 = – (x – 9)
2
3
27
3=– x+
2
2
3
33
y=– x+
2
2
Point-slope form
3
m = – , (𝑥1 , 𝑦1 ) = C(9, 3)
2
Distributive Property
Simplify.
Original equations
1
5
Substitute x + for y.
1
2
2
Subtract x from each side.
Subtract
2
33
2
from each side.
Divide each side by –2.
7
2
5
2
y = x + = (7) + = + = 6
The coordinates of the orthocenter of △ABC are (7, 6).
Exercises:
COORDINATE GEOMETRY Find the coordinates of the orthocenter of the triangle with the given vertices.
2. S(1, 0), T(4, 7), U(8, −3)
1. J(1, 0), H(6, 0), I(3, 6)
Chapter 5
12
Glencoe Geometry