Exercise 37 Page 640

Notice that the given triangles are both right triangles with a pair of congruent acute angles. Therefore, by the Angle-Angle Similarity Theorem, they are similar right triangles.

We are told that the area of the smaller triangle is $16{\text{ cm}}^2.$ If we calculate the area of the larger triangle, we will be able to find the area scale factor and therefore the scale factor. This will allow us to find $x$ and $y.$ To find the area of the larger triangle, we must recall that the area of a triangle is half the product of its base and its height. The area of the larger triangle is $48{\text{ cm}}^2.$ If the scale factor of the areas is $\frac{a^2}{b^2},$ then the ratio of corresponding side lengths is $\frac{a}{b}.$ With this in mind, we can find the scale factor. Be aware that in this context, a negative scale factor does not make sense. Therefore, we will only keep the principal root. Here, we can see that we have a radical in the denominator. Therefore, we must rationalize the denominator. To do this, we will multiply the numerator and denominator by $\sqrt{3}.$ We can use this information to find the values of $x$ and $y.$ Let's start by finding $x.$ Finally, we will find the value of $y$ by following the same procedure. We found that $x=\frac{8\sqrt{3}}{3}$ centimeters and $y=4\sqrt{3}$ centimeters.