sides of right angled triangle are 9,12,15.find sum of squares of medians

Question

 hypotenuse here is 15 

Answer 1

there is a formula that you can use

the median from point A=(1/2)√(2c^2+2b^2-a^2) where a, b, and c are the lengths of the sides opposite the angles A, B, and C

similarly the formula changes for each different median

you can use the formula or do the following...

draw right triangle ABC with A at the top, B on the right and C on the left

draw the medians

the median from C forms a right triangle with the median as the hypotenuse

call the median x

x^2=9^2+6^2  ( the median bisects AB so (1/2) of 12 is 6)

x^2=81+36=117 (the square of the first median)

the median from A forms a right triangle with the median as the hypotenuse

call the median y

y^2=12^2+4.5^2 (the median bisects BC so (1/2) of 9 is 4.5)

y^2=144+20.25=164.25 (the square of the second median)

the median from B intersects the hypotenuse of the original right triangle

the median to the hypotenuse of a right triangle is always half of the hypotenuse

call this median z

z=(1/2) of 15=7.5

z^2=7.5^2=56.25 (the square of the third median)

add the squares of the medians

117+164.25+56.25=337.5  answer

you can use the formula to find the three medians and square them to verify the answer

Answer 2

Draw and label a diagram.

Three internal triangles are formed.

One has legs of 9 and 6

One has legs of 4.5 and 12

One has legs of

 

Well you complete the pattern

Answer 3

The median = 1/2 the length of the 3rd side.

m₁ = 1/2(15) = 7 1/2

m₂ = 1/2(9) = 4 1/2

m₃ = 1/2(12) = 6

(7 1/2)² + (4 1/2)² + 6² = 112.5