in a right triangle, the hypothenuse is 10cm more than the shortest side. if third side is 6cm less than the hypotenuse, find the sides of triangle.
Answers
Let the base be p
Hypotenuse = 2p +6
Perpendicular = 2p + 4
By Pythagoras theoram
(2p+6)^2 = (2p+4)^2 +p^2
=> 4p^2 +36 + 24p = 4p^2 + 16 +16p +p^2
=> 36+ 24p = p^2 + 16p + 16
=> p^2 - 8p - 20 = 0
=> p^2 - 10p +2p - 20 = 0
=> p(p-10) +2(p-10) = 0
=> (p-10)(p+2) = 0
p = 10 and - 2
Length can't be negative
So,
p = 10
Base = 10
Perpendicular = 24
Hypotenuse = 26
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Let the RAT have its hypotenuse as H, and the other two sides as L and S,
H = 6+2S …(1)
L = H-2 or L = 6+2S-2 = 4+2S …(2)
H^2 = L^2+S^2, or
(6+2S)^2 = (4+2S)^2+S^2, or
36+24S+4S^2 = 16+16S+4S^2+S^2, or
S^2–8S-20=0 or,
S = [8+(64+80)^0.5]/2
= [8+12]/2 = 10
H = 6+2*10 = 26
L = 4+2S = 24
Check: H^2 = 26^2 = 676
L^2+S^2 = 24^2 + 10^2 = 576 + 100 = 676 = H^2. Correct.
The three sides of the triangle are 10, 24 and 26 units.