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 acute angle of \( 60^\circ \). This means the third angle is:

$$ 180^\circ - 90^\circ - 60^\circ = 30^\circ $$

This is a \( 30^\circ\text{-}60^\circ\text{-}90^\circ \) special right triangle.

Step 2: Apply the ratio rules

In a \( 30^\circ\text{-}60^\circ\text{-}90^\circ \) triangle, the side lengths are related as follows:

• Short leg (opposite \( 30^\circ \)) = \( s \)
• Long leg (opposite \( 60^\circ \)) = \( s\sqrt{3} \)
• Hypotenuse (opposite \( 90^\circ \)) = \( 2s \)

From the given image:

• The short leg is opposite the \( 30^\circ \) angle: \( s = 2\sqrt{3} \)
• The hypotenuse is \( x \):

$$ x = 2s = 2(2\sqrt{3}) = 4\sqrt{3} $$

• The long leg is \( y \):

$$ y = s\sqrt{3} = (2\sqrt{3})\sqrt{3} = 2 \cdot 3 = 6 $$

Step 3: Format the answers

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

• \( x = 4\sqrt{3} \) can be written as \( \sqrt{48} \) since \( 4\sqrt{3} = \sqrt{16 \cdot 3} = \sqrt{48} \). Following the format: sqrt(48)
• \( y = 6 \)

Answer:

x = sqrt(48)
y = 6