Question

find the length of the missing sides of the triangle. for any answers with a square root, simplify and write your answer in the form sqrt(...) with no spaces.

(m = \\)

(n = \\)

Explanation:

⚡ Using what you learned: special right triangles (45-45-90 and 30-60-90)

Step 1: Identify the triangle type

The given triangle is a right triangle with an angle of \( 45^\circ \). This makes it a \( 45^\circ\text{-}45^\circ\text{-}90^\circ \) special right triangle, where the legs are equal in length:

$$ m = n $$

The relationship between the legs and the hypotenuse is:

$$ \text{hypotenuse} = \text{leg} \cdot \sqrt{2} $$

Step 2: Set up the equation for the legs

Given the hypotenuse is \( 8 \):

$$ 8 = m\sqrt{2} $$

Solve for \( m \):

$$ m = \frac{8}{\sqrt{2}} $$

Step 3: Rationalize the denominator

$$ m = \frac{8}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2} $$

Since \( m = n \):

$$ n = 4\sqrt{2} $$

Step 4: Format the final answer

The prompt asks to write square root answers in the form sqrt(...) with no spaces:

$$ 4\sqrt{2} ightarrow 4\text{sqrt}(2) $$

Answer:

m = 4sqrt(2)
n = 4sqrt(2)