SOLUTION: In a right triangle, if the median from the right angle to
the hypotenuse is 5 in. what is the length of the hypotenuse?
I just want to make sure I am correct, the answer I cam
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Question 773066: In a right triangle, if the median from the right angle to
the hypotenuse is 5 in. what is the length of the hypotenuse?
I just want to make sure I am correct, the answer I came to was 21 using a^2+b^2=c^2.
Answer by pakhi(24) (Show Source): You can put this solution on YOUR website!
Let ABC be a right triangle , right angled at B. So AC is the hypotenuse.
Therefore by Pythagorus theorem AB^2 + BC^2 = AC^2 ----------(1)
Let D be the middle point of AC i.e. AD = CD ----------------(2)
BD is the the median from the right angle to the hypotenuse.
So BD = 5 in.
Also by Apollonius theorem we know that 'In any triangle , the sum of the squares on two sides is equal to twice the square on half the third side together with twice the square on the median that bisects the third side'.
Therefore in our case we can state that AB^2 + BC^2 = 2(AC/2)^2 + 2(BD)^2 --(3)
But AC/2 = AD -------(4)
or AC = 2AD --------(5)
Using equation 1 and equation 4 in equation 3 we have
AC^2 = 2(AD)^2 + 2(BD)^2 --------------(6)
Using equation 5 we can write equation 6 as
(2AD)^2 = 2(AD)^2 + 2(BD)^2
or 4(AD)^2 = 2(AD)^2 + 2(BD)^2
or 4(AD)^2 - 2(AD)^2 = 2(BD)^2
or 2(AD)^2 = 2(BD)^2
or AD^2 = BD^2
Putting the value of BD in the above equation we get
AD^2 = 5^2 = 25
Therefore AD = 5
From equation 5 we have AC = 2AD = 2*5 = 10
So the length of the hypotenuse is 10 in.
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