In a right triangle, the lengths of the 2 sides can be considered the base and height.
Also, in a right triangle: the sum of the squares of the length of sides is equal to the hypotenuse squared, this is called the pythagorean theorem.
Since both the base and height are in terms of x, we need to find x first in order to find area.
Therefore, using the pythagorean theorem we can find x.
Pythagorean Theorem: `a^2+b^2=c^2` , where a and b are the legs and c is the hypotenuse.
This gives us: `(5x-2)^2+(3x+1)^2=(5x)^2` Multiply out.
`(5x-2)^2 = (5x-2)(5x-2) = 25x^2 - 20x+4`
`(5x)^2 = (5x)(5x) = 25x^2`
Replacing these values into equation we have:
`25x^2-20x+4 + 9x^2+6x+1 = 25x^2` Combine like terms
`34x^2 -14x+5 = 25x^2` Since this is a quadratic, set the equation equal to 0, by subtracting `25x^2` from each side.
`9x^2 - 14x + 5 = 0` Now solve for x. Looks like you'll need to use the quadratic formula.
Factor to get: `(9x-5)(x-1)=0`
Therefore x = 1 or `5/9`
Now use these values to find the Area of the triangle
A = `1/2 bh`
A = `1/2(5x-2)(3x-1)`
A = `1/2(5•2.6817-2)(3•2.6817-1)`
A = .5(3)(4)
A = 6
A = `1/2(5•5/9-2)(3•5/9+1)`
A ≈ 1.037
Therefore, there are 2 possible answers: A = 6 or 1.037.