Question

1. (there are two right triangles in the image. the upper triangle has a right angle, one leg labeled w, the other leg labeled y, and the hypotenuse labeled $7\sqrt{2}$. the lower triangle has a right angle, one leg labeled w, the other leg labeled y, a hypotenuse labeled 7, and an acute angle of $45^\circ$.)

Explanation:

Step1: Identify triangle type

This is a 45-45-90 right triangle, so legs are equal: $w = y$.

Step2: Apply Pythagorean theorem

For right triangle: $w^2 + y^2 = 7^2$. Substitute $w=y$:
$$2w^2 = 49$$

Step3: Solve for $w$

$$w^2 = \frac{49}{2} \implies w = \sqrt{\frac{49}{2}} = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2}$$
Since $w=y$, $y = \frac{7\sqrt{2}}{2}$.

Step4: Verify top triangle (scaled version)

Top triangle hypotenuse is $7\sqrt{2}$, so legs are $\frac{7\sqrt{2}}{\sqrt{2}} = 7$, which matches the scaled relationship.

Answer:

For the lower triangle: $w = \frac{7\sqrt{2}}{2}$, $y = \frac{7\sqrt{2}}{2}$
For the upper triangle: $w = 7$, $y = 7$