Question

find the missing side lengths. for any answers with a square root, simplify and write your answer in the form sqrt(____) with no spaces.

\\x = \\
\\y = \\

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 \(45^\circ-45^\circ-90^\circ\) special right triangle, where the legs are equal in length:

$$x = y$$

Step 2: Set up the relationship

In a \(45^\circ-45^\circ-90^\circ\) triangle, the hypotenuse is \(\sqrt{2}\) times the length of a leg:

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

Given that the hypotenuse is \(4\):

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

Step 3: Solve for the leg length

$$x = \frac{4}{\sqrt{2}}$$

Rationalizing the denominator:

$$x = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$$

Since \(2 = \sqrt{4}\), we can write this in the required single square root format:

$$x = \sqrt{4 \cdot 2} = \sqrt{8}$$

Thus:

$$x = y = \sqrt{8}$$

Answer:

x = sqrt(8)
y = sqrt(8)