Answer
$\frac{4}{7x}$
Work Step by Step
Step 1: Find the hypotenuse of the right-angled triangle on the left.
Using the Pythagorean Theorem,
Hypotenuse = $\sqrt{(6x)^2+(8x)^2}=\sqrt{36x^2+64x^2}=\sqrt{100x^2}=10x$
Step 2: Find the hypotenuse of the right-angled triangle on the right.
Using the Pythagorean Theorem, Hypotenuse = $\sqrt{(15x)^2+(8x)^2}=\sqrt{225x^2+64x^2}=\sqrt{289x^2}=17x$
Step 3: Find the perimeter of the triangle.
The perimeter of the triangle is the sum of the length of all the sides of the triangle.
We have found that the triangle has sides of lengths $10x$, $17x$, and $(6x+15x)$.
Compute the perimeter: Perimeter = $10x+17x+21x=48x$
Step 4: Find the area of the triangle.
The height of the triangle is $8x$ and the base of the triangle is $(6x+15x)$.
We know the area of a triangle is $1/2\times base\times height$.
Using this formula,
Area of the triangle = $1/2\times 21x \times 8x=84x^2$
Step 5: Find the ratio of the perimeter to the area of the triangle.
Ratio $=\frac{48x}{84x^2}=\frac{12x\times 4}{12x\times 7x}=\frac{4}{7x}$
