Question

1. find the missing side lengths in the 45 - 45 - 90 triangles. leave your answers as radicals in simplest form.

9)
10)

1. find the missing side lengths in the 30 - 60 - 90 triangle. leave your answers as radicals in simplest form.

12)
13)
14)

Explanation:

Step1: Recall 30 - 60 - 90 and 45 - 45 - 90 triangle ratios

In 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 $c = a\sqrt{2}$. In 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 45 - 45 - 90 triangles

For a 45 - 45 - 90 triangle with hypotenuse $c=\sqrt{2}$, using $c = a\sqrt{2}$, we have $\sqrt{2}=a\sqrt{2}$, so $a = 1$. If one leg $a = 1$, the other leg is also 1.

Step3: Solve for 30 - 60 - 90 triangles

If the side opposite the 30 - degree angle in a 30 - 60 - 90 triangle is $a$, and the side opposite the 60 - degree angle is $y$ and hypotenuse is $x$. Given the side opposite 30 - degree angle $a$, then $y=a\sqrt{3}$ and $x = 2a$. If the side opposite 60 - degree angle is given as $y$, then $a=\frac{y}{\sqrt{3}}$ and $x=\frac{2y}{\sqrt{3}}$.

Answer:

The solutions depend on which specific triangle in the set is being referred to. For a 45 - 45 - 90 triangle with hypotenuse $\sqrt{2}$, the legs are 1. For a 30 - 60 - 90 triangle, if the side opposite 30 - degree angle is $a$, the side opposite 60 - degree angle is $a\sqrt{3}$ and hypotenuse is $2a$. If more specific details about which triangle number (7 - 14) are provided, more exact answers can be given.