Find the exact values of x and y. (x, y) = ( 60° 18 x y Submit Answer
The Correct Answer and Explanation is:
To determine the exact values of xxx and yyy, we need more context — such as a diagram, equation, or triangle reference — because “(x, y) = (60°, 18 x y)” is incomplete and ambiguous as it stands.
However, based on common geometry problems, let’s assume this is part of a triangle problem where one angle is 60∘60^\circ60∘, and the sides opposite to angles or adjacent to them are labeled xxx and yyy. One typical problem involves a 30°-60°-90° triangle, which has known side ratios.
Standard 30°-60°-90° Triangle:
In a 30°-60°-90° triangle, the side lengths are in a fixed ratio:
- Side opposite 30° is aaa
- Side opposite 60° is a3a\sqrt{3}a3
- Hypotenuse is 2a2a2a
Let’s assume:
- x=18x = 18x=18, which is the side opposite the 60° angle
- Then, we find the side opposite 30°, which is yyy
- And the hypotenuse, which would be 2y2y2y
From the triangle ratio, side opposite 60° is y3y\sqrt{3}y3
So:y3=18y\sqrt{3} = 18y3=18
Solve for yyy:y=183=1833=63y = \frac{18}{\sqrt{3}} = \frac{18\sqrt{3}}{3} = 6\sqrt{3}y=318=3183=63
So if:
- x=18x = 18x=18 (opposite 60°)
- y=63y = 6\sqrt{3}y=63 (opposite 30°)
The hypotenuse would be:2y=1232y = 12\sqrt{3}2y=123
Final Answer:
(x,y)=(18,63)(x, y) = (18, 6\sqrt{3})(x,y)=(18,63)
Explanation
This problem likely refers to a 30°-60°-90° triangle, a common type of right triangle in geometry with fixed ratios for its sides. The angles in this triangle are always 30 degrees, 60 degrees, and 90 degrees. The side lengths are always in the same proportion: the side opposite the 30° angle is the shortest, the side opposite the 60° angle is longer, and the hypotenuse is the longest.
Specifically, if the side opposite the 30° angle is represented by a variable aaa, then the side opposite the 60° angle is a3a\sqrt{3}a3, and the hypotenuse is 2a2a2a. This comes from basic trigonometric ratios in a right triangle and the geometric construction of this triangle from an equilateral triangle.
In the given case, if the side opposite the 60° angle is 18 units, then we set up the equation based on the known ratio:a3=18a\sqrt{3} = 18a3=18
Solving for aaa, which represents the side opposite 30°, we divide both sides by 3\sqrt{3}3:a=183=63a = \frac{18}{\sqrt{3}} = 6\sqrt{3}a=318=63
This value represents the side opposite the 30° angle. The hypotenuse, which is twice this value, would be 12312\sqrt{3}123. The triangle’s sides therefore perfectly match the expected proportions.
So, the exact values are x=18x = 18x=18, and y=63y = 6\sqrt{3}y=63, which maintain the geometric relationships of a 30°-60°-90° triangle.
