# The lengths of the sides of a right-angled triangle are (3x+1) cm, 5x cm, and (5x-2) cm. Find the area of the triangle. The 5x is the hypotenuse and 5x -2 is the base and 3x +1 is the height.

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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`

`(3x+1)^2= (3x+1)(3x+1)=9x^2+6x+1`

`(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

or

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.**

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In applying the theorem of Pythagorus to solve for `x` , whereafter the area of the triangle can be solved, we know that `x^2 +y^2 = h^2` , representing the squares of the two sides (base and height) which equal the square on the hypoteneuse (h). Care not to confuse the x in the equation with the x in the formula. Therefore :

`(3x+1)^2 + (5x-2)^2 = (5x)^2`

`(9x^2 +3x +3x +1) + (25x^2 -10x -10x + 4) = 25x^2`

`therefore 34x^2-14x +5 = 25x^2`

`therefore 9x^2-14x+5=0`

`therefore (x-1)(9x-5) = 0`

`therefore x=1 or x= 5/9`

Now use the formula to find the area of a right-angled triangle:

A=1/2bxh

We have the base: `(5x-2` ) and substituting when `x=1 = 5(1)-2= 3` and the height which is : `(3x+1) ` when` x=1 = 3(1)+1 = 4.` Therefore `A=1/2(3)(4` )

`therefore A=6` when x=1 and do the same when x=5/9

`therefore` `b=(5 (5/9)-2) = 7/9 and h=(3(5/9)+1)= 24/9`

`therefore A=1/2(7/9) x 24/9`

therefore A= 28/27 or 1.037. Remember to add the measurement dimensions to the answer (in this case centimeters)

**Ans: A=6 `cm^2` or A = 28/27 `cm^2` **