**How to Find the Slopes of Median and Altitude of a Triangle :**

Here we are going to see how to find the slopes of median and altitude of a triangle.

**Question 13 :**

A triangle has vertices at (6 , 7), (2 , -9) and (-4 , 1). Find the slopes of its medians.

**Solution :**

Median drawn through the vertex A will pass through the midpoint of the side BC.

Midpoint of BC = D

Midpoint = (x_{1} + x_{2})/2, (y_{1} + y_{2})/2

B(2, -9) and C(-4, 1)

= (2 - 4)/2, (-9 + 1)/2

= -2/2, -8/2

= D(-1, -4)

Slope of the median AD :

A(6, 7) D(-1, -4)

= (y_{2} - y_{1})/(x_{2} - x_{1})

= (-4 - 7) / (-1 - 6)

= -11/(-7)

= 11/7

Slope of the median AD = 11/7.

Median drawn through the vertex B will pass through the midpoint of the side AC.

Midpoint of AC = E

A(6, 7) and C(-4, 1)

= (6 - 4)/2, (7 + 1)/2

= 2/2, 8/2

= E(1, 4)

Slope of the median BE :

B(2, -9) E(1, 4)

= (y_{2} - y_{1})/(x_{2} - x_{1})

= (4 + 9)/(1 - 2)

= 13/(-1)

Slope of the median BE = -13

Median drawn through the vertex C will pass through the midpoint of the side AB.

Midpoint of AB = F

A(6 , 7) and B(2 , -9)

= (6 + 2)/2, (7 - 9)/2

= 8/2, -2/2

= F (4, -1)

Slope of the median CF :

C (-4 , 1) F (4, -1)

= (y_{2} - y_{1})/(x_{2} - x_{1})

= (-1 - 1)/(4 + 4)

= -2/8

Slope of the median CF = -1/4

**Question 14 :**

The vertices of a triangle ABC are A(-5 , 7), B(-4 , -5) and C(4,5). Find the slopes of the altitudes of the triangle.

**Solution :**

Every altitude drawn through the vertices of the triangle will be perpendicular to the other sides.

Slope of BC = (y_{2} - y_{1})/(x_{2} - x_{1})

= (5 + 5) / (4 + 4)

= 10/8

= 5/4

Slope of AD = -1/slope of BC

Slope of AD = -1 / (5/4) = -4/5

Slope of AC = (y_{2} - y_{1})/(x_{2} - x_{1})

= (5 - 7) / (4 + 5)

= -2/9

Slope of BE = -1/slope of AC

Slope of BE = -1 / (-2/9) = 9/2

Slope of AB = (y_{2} - y_{1})/(x_{2} - x_{1})

= (-5 - 7) / (-4 + 5)

= -12/1

= -12

Slope of CF = -1/slope of AB

Slope of CF = -1 /(-12) = 1/12

After having gone through the stuff given above, we hope that the students would have understood, how to find the slope of median and altitude of a triangle.

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