1. Medians and Altitudes of Triangles
Geometry
Mrs. Spitz

2. Objectives: Use properties of medians of a triangle
Use properties of altitudes of a triangle
8/1/2018
P. Spitz – Taos H.S.

3. Medians of a triangle
A median of a triangle is a segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. AD is a median in ∆ABC.
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4. Centroids of the Triangle
The three medians of a triangle are concurrent (they meet) at the CENTROID.
The centroid is
ALWAYS inside the
triangle.
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5. CENTROIDS -
ALWAYS INSIDE THE TRIANGLE
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6. So what?
The medians of a triangle have a special concurrency property.
The centroid of a triangle can be used as its balancing point. You already tried this yesterday .
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7. THEOREM Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point
that is two thirds of the distance from each
vertex to the midpoint of the opposite side.
If P is the centroid of ∆ABC, then
AP = 2/3 AD,
BP = 2/3 BF, and
CP = 2/3 CE
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8. Ex. 1: Using the Centroid of a Triangle
P is the centroid of ∆QRS shown below and
PT = 5. Find RT and RP.
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9. Because P is the centroid. RP = 2/3 RT.
Then PT= RT – RP = 1/3 RT. Substituting 5 for PT,
5 = 1/3 RT, so RT = 15.
Then RP = 2/3 RT = 2/3 (15) = 10
► So, RP = 10, and RT = 15.
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10. Ex. 2: Finding the Centroid of a Triangle
Find the coordinates of the centroid of ∆JKL
Choose the median KN. Find the coordinates of N, the midpoint of JL.
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11. Ex. 2: Finding the Centroid of a Triangle
The coordinates of N are:
3+7 , 6+10 = 10 , 16
Or (5, 8)
Find the distance from
vertex K to midpoint N.
The distance from K(5, 2)
to N (5, 8) is 8-2 or 6 units.
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12. Ex. 2: Finding the Centroid of a Triangle
Determine the coordinates of the centroid, which is 2/3 ∙ 6 or 4 units up from vertex K along median KN.
►The coordinates of centroid P are (5, 2+4), or (5, 6).
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13. 8/1/2018

14. Objective 2: Using altitudes of a triangle
An altitude of a triangle is the
perpendicular segment from the vertex to
the opposite side or to the line that
contains the opposite side. An altitude
can lie inside, on, or outside the triangle.
Every triangle has 3 altitudes. The lines
containing the altitudes are concurrent
and intersect at a point called the
orthocenter of the triangle.
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15. Ex. 3: Drawing Altitudes and Orthocenters
Where is the orthocenter located in each type of triangle?
Acute triangle
Right triangle
Obtuse triangle
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16. Acute Triangle - Orthocenter
∆ABC is an acute triangle.
The three altitudes intersect
at G, a point INSIDE the
triangle.
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17. Right Triangle - Orthocenter
∆KLM is a right triangle. The two
legs, LM and KM, are also altitudes.
They intersect at the triangle’s right
angle. This implies that the ortho
center is ON the triangle at M, the
vertex of the right angle of the
triangle.
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18. Obtuse Triangle - Orthocenter
∆YPR is an obtuse triangle. The three lines that contain the altitudes intersect at W, a point that is OUTSIDE the triangle.
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