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.

\\(x =\\)
\\(y =\\)

Explanation:

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

Step 1: Identify the triangle type and side relationships

The given triangle is a right triangle with an acute angle of \( 30^\circ \), making it a \( 30^\circ\text{-}60^\circ\text{-}90^\circ \) special right triangle.

The ratios of the sides opposite to the angles \( 30^\circ : 60^\circ : 90^\circ \) are:

$$ 1 : \sqrt{3} : 2 $$

Let:

• The shorter leg (opposite \( 30^\circ \)) be \( y \)
• The longer leg (opposite \( 60^\circ \)) be \( x \)
• The hypotenuse (opposite \( 90^\circ \)) be \( 56 \)

Step 2: Calculate the shorter leg \( y \)

Using the hypotenuse relationship:

$$ \text{hypotenuse} = 2y $$

$$ 56 = 2y $$

$$ y = 28 $$

Step 3: Calculate the longer leg \( x \)

Using the relationship between the legs:

$$ x = y\sqrt{3} $$

$$ x = 28\sqrt{3} $$

Formatting the square root as requested: 28sqrt(3)

Answer:

• \( x = \) 28sqrt(3)
• \( y = \) 28