Math  /  Geometry

QuestionFind the value of yy 4 232 \sqrt{3} 8×38 \times 3 8

Studdy Solution

STEP 1

1. We are given a right triangle.
2. The triangle has angles 3030^\circ, 6060^\circ, and 9090^\circ.
3. The side opposite the 3030^\circ angle is 434\sqrt{3}.
4. We need to find the side opposite the 6060^\circ angle, labeled yy.

STEP 2

1. Identify the properties of a 3030^\circ, 6060^\circ, 9090^\circ triangle.
2. Use the properties to find the length of the side opposite the 6060^\circ angle.
3. Verify the solution with the given answer choices.

STEP 3

In a 3030^\circ, 6060^\circ, 9090^\circ triangle, the sides are in the ratio 1:3:21 : \sqrt{3} : 2. The side opposite the 3030^\circ angle is the shortest and is labeled as xx, the side opposite the 6060^\circ angle is x3x\sqrt{3}, and the hypotenuse is 2x2x.

STEP 4

Given that the side opposite the 3030^\circ angle is 434\sqrt{3}, we can set up the equation:
x=43 x = 4\sqrt{3}

STEP 5

Using the properties of the triangle, the side opposite the 6060^\circ angle is x3x\sqrt{3}. Substitute x=43x = 4\sqrt{3} into the equation:
y=x3=(43)3 y = x\sqrt{3} = (4\sqrt{3})\sqrt{3}

STEP 6

Simplify the expression:
y=43×3=4×3=12 y = 4\sqrt{3} \times \sqrt{3} = 4 \times 3 = 12

STEP 7

Verify the solution with the given answer choices: 4, 232\sqrt{3}, 838\sqrt{3}, and 8. The calculated value 1212 is not among the given choices, indicating a potential error in the problem statement or answer choices.
The value of yy is calculated as:
12 \boxed{12}

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