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Solve for x and y in a 30-60-90 Right Triangle with Side Length 18
Mathematics
Grade 10 (Junior High School)
Question Content
Find the values of x and y in the right triangle with one acute angle of 60°, the side opposite the 30° angle is 18. If necessary, write answers in radical form.
Correct Answer
x = 18√3, y = 36
Detailed Solution Steps
1
Step 1: Identify the angles of the right triangle. Since it is a right triangle (90°) with one 60° angle, the third angle is 180° - 90° - 60° = 30°.
2
Step 2: Relate the sides to the 30-60-90 special right triangle properties. In a 30-60-90 triangle, the side opposite 30° is the shortest side (let's call it s), the side opposite 60° is s√3, and the hypotenuse is 2s.
3
Step 3: Match the given side to the correct angle. The side with length 18 is opposite the 30° angle, so s = 18.
4
Step 4: Calculate x (side opposite 60°): x = s√3 = 18√3.
5
Step 5: Calculate y (the hypotenuse, opposite 90°): y = 2s = 2*18 = 36.
Knowledge Points Involved
1
30-60-90 Special Right Triangle Properties
A right triangle with angles 30°, 60°, 90° has a fixed side ratio: shortest side (opposite 30°) : side opposite 60° : hypotenuse = 1 : √3 : 2. This allows quick calculation of side lengths if one side is known, without using trigonometric functions.
2
Triangle Angle Sum Theorem
The sum of the interior angles of any triangle is 180°. For right triangles, this means the two acute angles add up to 90°, which helps identify the measure of the unknown acute angle.
3
Trigonometric Ratios (Alternative Method)
Using tangent and secant (or cosine) ratios: tan(60°) = opposite/adjacent = x/18, so x = 18*tan(60°) = 18√3; cos(60°) = adjacent/hypotenuse = 18/y, so y = 18/cos(60°) = 18/0.5 = 36. This confirms the result from the special triangle properties.
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