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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 โณ ABC , U is the centroid and
BU = 16. Find UK and BK****.
BU =
2 3 BK
16 = 2 3
BK
24 = BK
BU + UK = BK
16 + UK = 24
UK = 8
Exercises:
In โณ ABC , AU = 16, BU = 12, and CF = 18. Find each measure.
1. UD 2. EU
3. CU 4. AD
5. UF 6. BE
In โณ 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
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 โณ ABC are A (1, 3),
B (7, 7) and C (9, 3). Find the coordinates of the orthocenter of โณ ABC****.
Find the point where two of the three altitudes intersect.
Find the equation of the altitude from A to ๐ต๐ถฬ
ฬ
ฬ
ฬ
.
If ๐ต๐ถฬ
ฬ
ฬ
ฬ
has a slope of โ2, then the altitude
has a slope of
1
y โ ๐ฆ 1 = m ( x โ ๐ฅ 1 ) Point-slope form
y โ 3 = 1 2 ( x โ 1) m = 12 , (๐ฅ 1 , ๐ฆ 1 ) = A (1, 3)
y โ 3 =
1 2 x^ โ^
1 2 Distributive Property y =
1 2 x +^
5 2 Simplify.
Find the equation of the altitude from C to ๐ด๐ต ฬ
ฬ
ฬ
ฬ
ฬ
.
If ๐ด๐ตฬ
ฬ
ฬ
ฬ
has a slope of 2 3 , then the altitude has a slope
of โ 3 2
y โ ๐ฆ 1 = m ( x โ ๐ฅ 1 ) Point-slope form
y โ 3 = โ
3 2 ( x^ โ^ 9)^ m^ =^ โ^
3 2 , (๐ฅ^1 ,^ ๐ฆ^1 ) =^ C (9, 3) y โ 3 = โ 3 2 x + 27 2 Distributive Property
y = โ
3 2 x^ +^
33 2 Simplify.
Solve the system of equations and find where the altitudes meet.
y = 1 2 x + 5 2 y = โ 3 2 x + 33 2 Original equations 1 2 x + 5 2
3 2 x + 33 2 Substitute 12 x + 52 for y. 5 2 = โ2 x^ +^
33 2 Subtract^
1 2 x^ from each side. โ14 = โ2 x Subtract 332 from each side.
7 = x Divide each side by โ 2.
y = 1 2 x + 5 2
1 2
5 2
7 2
5 2
The coordinates of the orthocenter of โณ ABC 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 (8, โ3)
