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.
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.
answer plz
but check for typos
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.
Never mind, it has nice whole real roots
To find the length of the shortest side, we can use the Pythagorean theorem and the formula to calculate the area of a triangle.
Let's denote the lengths of the two shorter sides as x and x+3 (since they differ by 3cm). The longest side (the hypotenuse) can be denoted as √(x^2 + (x+3)^2), by applying the Pythagorean theorem.
According to the formula for the area of a triangle, Area = (1/2) * base * height, where base and height are two sides of the triangle.
Since the two shorter sides are the base and height of the triangle, we can use the given area to represent the equation:
104 = (1/2) * x * (x+3)
Now, let's solve this equation to find the value of x:
104 = (1/2) * x * (x+3)
208 = x^2 + 3x
x^2 + 3x - 208 = 0
Now, we can solve this quadratic equation to find the value of x. We can either factor it or use the quadratic formula:
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = 3, and c = -208.
x = (-3 ± √(3^2 - 4(1)(-208))) / 2(1)
x = (-3 ± √(9 + 832)) / 2
x = (-3 ± √841) / 2
x = (-3 ± 29) / 2
Therefore, we have two solutions:
1. x = (-3 + 29) / 2 = 13
2. x = (-3 - 29) / 2 = -16
Since the length of a side cannot be negative, the shortest side of the triangle is 13 cm.