Question

regular
date 10/09/2025 period 4
find the missing side lengths. round answers to the 100ths place.
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find the missing side lengths. leave your answers as radicals in simplest form.
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Explanation:

Step1: Identify triangle - type

These are 45 - 45 - 90 and 30 - 60 - 90 right - triangles. Use their side - length ratios.
For a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$. For a 30 - 60 - 90 triangle, the ratio of the sides is $1:\sqrt{3}:2$.

Step2: Solve for missing sides in 45 - 45 - 90 triangles

If the length of one of the legs is $a$, the other leg has the same length $a$, and the hypotenuse $c=a\sqrt{2}$. For example, if the hypotenuse $c$ is given, then the leg length $a = \frac{c}{\sqrt{2}}$.

Step3: Solve for missing sides in 30 - 60 - 90 triangles

If the shorter leg (opposite the 30 - degree angle) has length $a$, the longer leg (opposite the 60 - degree angle) has length $a\sqrt{3}$, and the hypotenuse has length $2a$. If the longer leg is given as $b$, then the shorter leg $a=\frac{b}{\sqrt{3}}$, and if the hypotenuse $c$ is given, the shorter leg $a=\frac{c}{2}$ and the longer leg $b = \frac{c\sqrt{3}}{2}$.

Answer:

The work shown in the image already contains the correct answers for finding the missing side - lengths of the right - triangles. For example:

1. In the first 45 - 45 - 90 triangle, if the hypotenuse is $3\sqrt{2}$, then each leg is $\frac{3\sqrt{2}}{\sqrt{2}} = 3$.
2. In the second 45 - 45 - 90 triangle, if one leg is $\frac{5\sqrt{2}}{3}$, the other leg is also $\frac{5\sqrt{2}}{3}$.
3. In the 30 - 60 - 90 triangle, if the shorter leg is $\frac{3}{2}$, the longer leg is $\frac{3\sqrt{3}}{2}$ and if the longer leg is given and we need to find the shorter leg or hypotenuse, we use the appropriate ratio.
4. In the fourth 30 - 60 - 90 triangle, if the hypotenuse is $5\sqrt{3}$, the shorter leg is $\frac{5\sqrt{3}}{2}$ and the longer leg is $\frac{5\sqrt{3}\times\sqrt{3}}{2}=\frac{15}{2}$.
5. In the fifth 30 - 60 - 90 triangle, if the longer leg is 2, the shorter leg is $\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$.
6. In the sixth 45 - 45 - 90 triangle, if the hypotenuse is $2\sqrt{5}$, each leg is $\frac{2\sqrt{5}}{\sqrt{2}}=\sqrt{10}$.
7. In the seventh 45 - 45 - 90 triangle, if the hypotenuse is $\frac{7\sqrt{20}}{21}=\frac{7\times2\sqrt{5}}{21}=\frac{2\sqrt{5}}{3}$, each leg is $\frac{\frac{2\sqrt{5}}{3}}{\sqrt{2}}=\frac{\sqrt{10}}{3}$.
8. In the eighth 30 - 60 - 90 triangle, if the shorter leg is $\frac{4\sqrt{3}}{5}$, the longer leg is $\frac{4\sqrt{3}}{5}\times\sqrt{3}=\frac{12}{5}$.
9. For the 30 - 60 - 90 triangle with hypotenuse $2\sqrt{3}$, the shorter leg is $\sqrt{3}$ and the longer leg is 3.
10. For the 45 - 45 - 90 triangle with leg $\frac{9}{2}$, the other leg is $\frac{9}{2}$ and the hypotenuse is $\frac{9\sqrt{2}}{2}$.