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Calculate x, y, z in composite 30-60-90 and 45-45-90 right triangles with hypotenuse 12
Mathematics
Grade 10 (Junior High)
Question Content
Solve for x, y, z in a composite right triangle with angle 30°, hypotenuse 12, attached to a 45-45-90 right triangle.
Correct Answer
x=6, y=6√3, z=6√2
Detailed Solution Steps
1
Step 1: For the 30-60-90 triangle with hypotenuse 12, side x (opposite 30°) = 12/2=6. Side y (opposite 60°)=6×√3=6√3.
2
Step 2: Side x is the leg of the 45-45-90 triangle, so hypotenuse z=6×√2=6√2.
Knowledge Points Involved
1
30-60-90 Special Right Triangle Properties
In a right triangle with angles 30°, 60°, 90°, the side lengths follow the ratio 1 : √3 : 2, where the shortest side (opposite 30°) is half the hypotenuse, and the longer leg (opposite 60°) is the shortest side multiplied by √3. Used to quickly calculate side lengths without trigonometric functions.
2
45-45-90 Special Right Triangle Properties
In a right triangle with two 45° angles, the triangle is isosceles, so the two legs are equal, and the hypotenuse is leg length × √2. This allows quick calculation of side lengths for isosceles right triangles.
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