The problem asks us to find the value of $y$ in two right triangles. a) We have a right triangle with a $45^\circ$ angle, a base of length 1, and a height of length 1. $y$ is the hypotenuse. b) We have a right triangle with a $60^\circ$ angle, a base of length 1, and a hypotenuse of length 2. $y$ is the height.

1. Problem Description

The problem asks us to find the value of $y$ in two right triangles.

a) We have a right triangle with a $45^\circ$ angle, a base of length 1, and a height of length

1. $y$ is the hypotenuse.

b) We have a right triangle with a $60^\circ$ angle, a base of length 1, and a hypotenuse of length

2. $y$ is the height.

2. Solution Steps

a) In the first triangle, we can use the Pythagorean theorem to find $y$.

$y^2 = 1^2 + 1^2$

$y^2 = 1 + 1$

$y^2 = 2$

$y = \sqrt{2}$

Alternatively, since we know that the triangle is a $45-45-90$ triangle, and the legs are 1, then the hypotenuse is $\sqrt{2}$.

b) In the second triangle, we can use the Pythagorean theorem to find $y$. We have $1^2 + y^2 = 2^2$

$1 + y^2 = 4$

$y^2 = 4 - 1$

$y^2 = 3$

$y = \sqrt{3}$

Alternatively, we can use trigonometry. In a $30-60-90$ triangle, the side opposite the $60^\circ$ angle is $\sqrt{3}$ times the length of the side opposite the $30^\circ$ angle. Since the side opposite the $30^\circ$ angle has length 1, the side opposite the $60^\circ$ angle has length $\sqrt{3}$.

3. Final Answer

a) $y = \sqrt{2}$

b) $y = \sqrt{3}$