Question

Find the missing side lengths. Leave your answers as radicald in simplest forms.​

Answer 1

x = 10

y = 5

Explanation:

Interior angles of a triangle sum to 180°. Therefore, from inspection of the given diagram, the interior angles of the given right triangle are:

• 30°, 60° and 90°

This means the triangle is a special right triangle and that the 30-60-90 theorem can be applied to quickly find the measures of the missing sides.

30-60-90 Triangle Theorem

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio 1 : √3 : 2.

Therefore, the formula for the ratio of the sides is b : b√3 : 2b where:

• b = the shortest side opposite the 30° angle.
• b√3 = the side opposite the 60° angle.
• 2b = the longest side (hypotenuse) opposite the right angle.

From inspection of the given triangle, the side opposite the 60° angle is 5√3. Therefore:

⇒ b√3 = 5√3

⇒ b = 5

Substitute the found value of b into the expressions for the other two sides to find the values of x and y:

⇒ x = 2b = 2 · 5 = 10

⇒ y = b = 5