AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Find x, y, z in composite 45-45-90 and 30-60-90 right triangles with side 32
Mathematics
Grade 10 (Junior High)
Question Content
Solve for x, y, z in a composite right triangle with angles 45°, 60°, right angle, and side 32.
Correct Answer
x=32√2, y=16√6, z=16√2
Detailed Solution Steps
1
Step 1: First, look at the 45-45-90 triangle with leg 32: hypotenuse x=32×√2=32√2.
2
Step 2: Now look at the 30-60-90 triangle where x is the hypotenuse. The side z is opposite 30°, so z=32√2/2=16√2.
3
Step 3: The side y is opposite 60°, so y=16√2×√3=16√6.
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...
