Question

find the missing side lengths. round answers to the 100ths place.
find the missing side lengths. leave your answers as radicals in simplest form.

Explanation:

Step1: Identify right - triangle trigonometry relationships

For a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$, i.e., if the legs are of length $a$, the hypotenuse $c = a\sqrt{2}$. For a 30 - 60 - 90 triangle, the ratio of the sides is $1:\sqrt{3}:2$, where the side opposite the 30 - degree angle is $a$, the side opposite the 60 - degree angle is $a\sqrt{3}$, and the hypotenuse is $2a$.

Step2: Solve for side lengths in each triangle

For example, in a 30 - 60 - 90 triangle, if the side opposite the 60 - degree angle is $5\sqrt{3}$, and we know that if the side opposite 30 - degree angle is $x$ and the side opposite 60 - degree angle is $y$, then $y = x\sqrt{3}$. If $y = 5\sqrt{3}$, then $x = 5$.

Step3: Round answers as required

When the problem asks to round to the hundredths place, calculate the decimal value of the radical expressions. For example, $\sqrt{10}\approx3.16$.

Answer:

The solutions for the missing side - lengths are as follows (rounded to the hundredths place where required):

1. $a = 3$, $b\approx2.12$
2. $x\approx2.36$, $y\approx2.36$
3. $x = 3$, $y\approx2.60$
4. $x = 5$, $y\approx4.33$
5. $x\approx1.15$, $y = 2$
6. $x\approx3.16$
7. $u=\frac{2\sqrt{10}}{3}$, $v=\frac{7\sqrt{20}}{21}$
8. $x=\frac{4}{5}$

(Note: The last two triangles in the image (9 and 10) are incomplete in the provided image, so no solutions are given for them.)