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Math  /  Geometry

Question7. x=x= \qquad y=y=

Studdy Solution

STEP 1

1. We have a right triangle.
2. The hypotenuse is 46 46 .
3. One angle is 60∘ 60^\circ .
4. We need to find the lengths of the sides x x and y y .

STEP 2

1. Identify the type of triangle and the relationships between its sides.
2. Use trigonometric ratios to find the lengths of x x and y y .

STEP 3

Identify the type of triangle. We have a right triangle with one angle measuring 60∘ 60^\circ . The other non-right angle must be 30∘ 30^\circ because the sum of angles in a triangle is 180∘ 180^\circ . This is a 30-60-90 triangle, which is a special right triangle.

STEP 4

In a 30-60-90 triangle, the sides are in the ratio 1:3:2 1 : \sqrt{3} : 2 . The side opposite the 30∘ 30^\circ angle is the shortest and is half the hypotenuse. The side opposite the 60∘ 60^\circ angle is the longest leg and is 3 \sqrt{3} times the shortest side.

STEP 5

Since the hypotenuse is 46 46 , the side opposite the 30∘ 30^\circ angle (shortest side) is:
x=462=23 x = \frac{46}{2} = 23

STEP 6

The side opposite the 60∘ 60^\circ angle (longest leg) is:
y=23×3 y = 23 \times \sqrt{3}
The values of x x and y y are:
x=23 x = 23 y=233 y = 23\sqrt{3}

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