Question

1. find the missing side lengths using special triangle rules a) $x = \frac{5\sqrt{2}}{2}, y = 5$ b) $x = 5\sqrt{2}, y = \frac{5\sqrt{2}}{2}$ c) $x = \frac{5\sqrt{2}}{2}, y = \frac{5\sqrt{2}}{2}$ d) $x = 5\sqrt{2}, y = 5$ 9. find the missing side lengths using special triangle rules. a) $x = 2\sqrt{3}, y = 2\sqrt{2}$ b) $x = 2\sqrt{2}, y = 2\sqrt{3}$ c) $x = 2\sqrt{2}, y = 2$ d) $x = \sqrt{2}, y = 2\sqrt{3}$

Explanation:

Step1: Identify the triangle type

The given triangles are 45 - 45 - 90 special right - triangles. In a 45 - 45 - 90 triangle, the ratio of the side lengths is $a:a:a\sqrt{2}$, where the legs have length $a$ and the hypotenuse has length $a\sqrt{2}$.

Step2: Solve for problem 8

In the first triangle, one leg has length 5. Since the two legs of a 45 - 45 - 90 triangle are equal, $y = 5$. The hypotenuse $x$ is given by the formula $x=a\sqrt{2}$, with $a = 5$, so $x = 5\sqrt{2}$.

Step3: Solve for problem 9

In the second triangle, one leg has length 2. Since the two legs of a 45 - 45 - 90 triangle are equal, $y = 2$. The hypotenuse $x$ is given by the formula $x=a\sqrt{2}$, with $a = 2$, so $x=2\sqrt{2}$.

Answer:

1. d) $x = 5\sqrt{2}, y = 5$
2. c) $x = 2\sqrt{2}, y = 2$