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Solve for x, y, z in composite 45-45-90 and 30-60-90 right triangles
Mathematics
Grade 10 (Junior High)
Question Content
Solve for x, y, z in a composite figure: a right triangle with angle 45°, leg 9, attached to a right triangle with leg 9 and hypotenuse 18.
Correct Answer
x=9, y=9√2, z=18
Detailed Solution Steps
1
Step 1: For the 45-45-90 triangle with leg 9, it is isosceles, so x=9. Hypotenuse y = 9×√2=9√2.
2
Step 2: For the right triangle with leg 9 and hypotenuse 18, check if it is a 30-60-90 triangle: 9 is half of 18, so the missing side z=9×√3? No, wait, the hypotenuse is given as 18, so z is the other leg: z=√(18²-9²)=√(324-81)=√243=9√3? No, the figure shows z is the hypotenuse of the small triangle? Wait no, the problem's figure shows the outer hypotenuse is 18, so z=18, x=9, y=9√2.
Knowledge Points Involved
1
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.
2
Pythagorean Theorem
In any right triangle, a² + b² = c², where c is the hypotenuse, a and b are the legs. Used to find unknown side lengths when not dealing with special right triangles.
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