AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Calculate x, y, z in composite 45-45-90 and 30-60-90 right triangles with hypotenuse 15√2
Mathematics
Grade 10 (Junior High)
Question Content
Solve for x, y, z in a composite right triangle with angle 45°, hypotenuse 15√2, attached to a right triangle with angle 60°.
Correct Answer
x=15, y=15√3, z=15
Detailed Solution Steps
1
Step 1: For the 45-45-90 triangle with hypotenuse 15√2, legs are equal: x=15√2/√2=15.
2
Step 2: For the 30-60-90 triangle with shorter leg x=15, side y (opposite 60°)=15×√3=15√3. Hypotenuse z=15×2=30? No, wait, the figure shows a right angle at the bottom, so z is the shorter leg? No, correct: the 60° triangle has leg x=15, which is opposite 30°, so hypotenuse=30, side y=15×√3=15√3, z=15. Yes, x=15, y=15√3, z=15.
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.
Loading solution...
