# mathematics

the length of two shorter sides of a right angled triangle differ by 3cm .if the area of the triangle is 104cm.find the length of the shortest side.

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2. shorter side ---- x cm
longer side = x+3 cm

area of triangle = (1/2)(base)(height)
(1/2)(x)(x+3) = 104
x(x+3) = 208
x^2 + 3x - 208 = 0

I assume you know how to solve a quadratic equation, since these
kind of problems arise in the study of the quadratics.

hint: well, what do you know, it even factors.

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3. but check for typos

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4. Never mind, it has nice whole real roots

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5. Since it says the "two shorter sides", we know the hypotenuse is not involved, since it's always the longest side.

From here, let's call the shortest side 'x', and thus, the longer side will be x+3.

For triangles, A = (1/2)(bh), so now just sub. in what we know:

104 = (1/2)(x)(x+3)
104 = (1/2)(x²+3x)
208 = x²+3x
0 = x²+3x-208 ---> Now, factor this:

0 = (x-13)(x+16)
x = 13, -16

Obviously, a side length cannot be negative, so -16 is extraneous.

Since we defined x as the shortest side, and it was found to be 13, the length of the shortest side is 13 cm.

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