Question

How many right triangles have a hypotenuse that measures 2x + 3 meters and legs that measure 2x - 5 meters and x + 7 meters? What are their dimensions?

Answer 1

Two right angled triangle are possible.

Explanation:

We are given the following in the question:

Measure of hypotenuse in meters =
`2x + 3`

Measure of legs in meters =
`2x -5 \text{ and }x+7`

In a hypotenuse triangle:

`(\text{Hypotenuse})^2 = (\text{leg}_1)^2 + (\text{leg}_2)^2`

Putting the values, we get,

`(2x+3)^2 = (2x-5)^2 + (x+7)^2\\4x^2 + 9+12x = (4x^2 + 25 - 20x) + (x^2 + 49 +14x)\\4x^2 + 9 +12x = 5x^2 -6x +74\\x^2-18x +65 = 0\\x^2 - 13x - 5x + 65 = 0\\x(x-13)-5(x-13) = 0\\(x-13)(x-5) = 0\\x = 13, x = 5`

Thus, there are two triangles possible with dimension:

Triangle 1:

`\text{Hypotenuse} = 2x + 3 = 2(13) + 3 = 29\\\text{leg}_1 = 2x-5 = 2(13) - 5 = 21\\\text{leg}_2 = x + 7 = 13 + 7 = 20\\`

Triangle 2:

`\text{Hypotenuse} = 2x + 3 = 2(5) + 3 = 13\\\text{leg}_1 = 2x-5 = 2(5) - 5 = 5\\\text{leg}_2 = x + 7 = 5 + 7 = 12\\`