Question

practice 45,45,90
regular
find the missing side lengths. leave your answers as radicals in simplest form. no decimals.

1. find the missing side lengths. round answers to the 100th place.

Explanation:

Step1: Recall 45 - 45-90 triangle ratio

In a 45 - 45 - 90 triangle, the ratio of the side lengths is $a:b:c = 1:1:\sqrt{2}$, where $a$ and $b$ are the legs and $c$ is the hypotenuse. That is, if the length of a leg is $x$, the length of the other leg is also $x$, and the length of the hypotenuse is $x\sqrt{2}$.

Step2: Solve for missing sides in each triangle

For example, if the hypotenuse $c$ is given, and we know $c = x\sqrt{2}$, then $x=\frac{c}{\sqrt{2}}=\frac{c\sqrt{2}}{2}$. If a leg $x$ is given, and we want to find the hypotenuse $c$, then $c = x\sqrt{2}$.

1. If one leg is 5, then the other leg $b = 5$ and the hypotenuse $a=5\sqrt{2}$ (since $a$ is the hypotenuse and using $c = x\sqrt{2}$ with $x = 5$).
2. If the hypotenuse is 10, then a leg $x=\frac{10}{\sqrt{2}}=\frac{10\sqrt{2}}{2}=5\sqrt{2}$.
3. If the hypotenuse is $2\sqrt{2}$, then a leg $u=\frac{2\sqrt{2}}{\sqrt{2}} = 2$.
4. If a leg is $\frac{2\sqrt{6}}{3}$, and we know the ratio of leg to hypotenuse is $1:\sqrt{2}$, then if the leg is $x=\frac{2\sqrt{6}}{3}$, and we want to find the other leg $b$, since legs are equal in 45 - 45 - 90 triangle, and if we use the ratio to find the hypotenuse - related value, $b=\frac{4\sqrt{3}}{3}$ (using the ratio and simplification of the equations based on the 45 - 45 - 90 triangle properties).
5. If a leg is $\frac{3\sqrt{2}}{2}$, then the other leg $b = 3$ (using the ratio and cross - multiplication: $\frac{b}{\frac{3\sqrt{2}}{2}}=\frac{\sqrt{2}}{1}$, so $b = 3$).
6. If a leg is 1, then the other leg $u = 1$ and the hypotenuse is $\sqrt{2}$.
7. If a leg is $\frac{4\sqrt{2}}{5}$, then the other leg $y=\frac{4}{5}=0.8$ (using the ratio of the legs being equal).
8. If the hypotenuse is $8\sqrt{13}$, then a leg $b=\frac{8\sqrt{13}}{\sqrt{2}}=\frac{8\sqrt{26}}{2}=4\sqrt{26}\approx4\times5.099 = 20.396\approx20.40$.
9. If a leg is $\frac{7\sqrt{2}}{2}$, then the other leg $n = 7$ (using the ratio $\frac{n}{\frac{7\sqrt{2}}{2}}=\frac{\sqrt{2}}{1}$, cross - multiply to get $n = 7$).
10. If a leg is 8, then the other leg $b = 8\sqrt{2}\approx8\times1.414 = 11.312\approx11.31$.

The solutions for each triangle's missing side lengths are as shown in the hand - written work based on the 45 - 45 - 90 triangle side - length ratio properties.

Answer:

Step1: Recall 45 - 45-90 triangle ratio

In a 45 - 45 - 90 triangle, the ratio of the side lengths is $a:b:c = 1:1:\sqrt{2}$, where $a$ and $b$ are the legs and $c$ is the hypotenuse. That is, if the length of a leg is $x$, the length of the other leg is also $x$, and the length of the hypotenuse is $x\sqrt{2}$.

Step2: Solve for missing sides in each triangle

For example, if the hypotenuse $c$ is given, and we know $c = x\sqrt{2}$, then $x=\frac{c}{\sqrt{2}}=\frac{c\sqrt{2}}{2}$. If a leg $x$ is given, and we want to find the hypotenuse $c$, then $c = x\sqrt{2}$.

1. If one leg is 5, then the other leg $b = 5$ and the hypotenuse $a=5\sqrt{2}$ (since $a$ is the hypotenuse and using $c = x\sqrt{2}$ with $x = 5$).
2. If the hypotenuse is 10, then a leg $x=\frac{10}{\sqrt{2}}=\frac{10\sqrt{2}}{2}=5\sqrt{2}$.
3. If the hypotenuse is $2\sqrt{2}$, then a leg $u=\frac{2\sqrt{2}}{\sqrt{2}} = 2$.
4. If a leg is $\frac{2\sqrt{6}}{3}$, and we know the ratio of leg to hypotenuse is $1:\sqrt{2}$, then if the leg is $x=\frac{2\sqrt{6}}{3}$, and we want to find the other leg $b$, since legs are equal in 45 - 45 - 90 triangle, and if we use the ratio to find the hypotenuse - related value, $b=\frac{4\sqrt{3}}{3}$ (using the ratio and simplification of the equations based on the 45 - 45 - 90 triangle properties).
5. If a leg is $\frac{3\sqrt{2}}{2}$, then the other leg $b = 3$ (using the ratio and cross - multiplication: $\frac{b}{\frac{3\sqrt{2}}{2}}=\frac{\sqrt{2}}{1}$, so $b = 3$).
6. If a leg is 1, then the other leg $u = 1$ and the hypotenuse is $\sqrt{2}$.
7. If a leg is $\frac{4\sqrt{2}}{5}$, then the other leg $y=\frac{4}{5}=0.8$ (using the ratio of the legs being equal).
8. If the hypotenuse is $8\sqrt{13}$, then a leg $b=\frac{8\sqrt{13}}{\sqrt{2}}=\frac{8\sqrt{26}}{2}=4\sqrt{26}\approx4\times5.099 = 20.396\approx20.40$.
9. If a leg is $\frac{7\sqrt{2}}{2}$, then the other leg $n = 7$ (using the ratio $\frac{n}{\frac{7\sqrt{2}}{2}}=\frac{\sqrt{2}}{1}$, cross - multiply to get $n = 7$).
10. If a leg is 8, then the other leg $b = 8\sqrt{2}\approx8\times1.414 = 11.312\approx11.31$.

The solutions for each triangle's missing side lengths are as shown in the hand - written work based on the 45 - 45 - 90 triangle side - length ratio properties.