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Solve for Hypotenuse x and Leg y in a 45-45-90 Right Triangle
Mathematics
Grade 8 (Junior High School)
Question Content
In a 45-45-90 right triangle, find the values of x (hypotenuse) and y (leg). No side length is given, but the handwritten answer shows x = 8, y = 8√2 (assume the leg length is 8, matching the context of question 1).
Correct Answer
x = 8√2, y = 8
Detailed Solution Steps
1
Step 1: Identify the triangle type. This is a 45-45-90 right isosceles triangle, with side ratio leg : leg : hypotenuse = 1 : 1 : √2.
2
Step 2: Assume the leg length y is 8 (consistent with the adjacent problem context). Since the legs are congruent, y = 8.
3
Step 3: Calculate the hypotenuse x. Use the ratio hypotenuse = leg × √2, so x = 8 × √2 = 8√2. (Note: The handwritten answer has x and y reversed; the correct assignment matches the side labels in the triangle.)
Knowledge Points Involved
1
45-45-90 Special Right Triangle Side Ratio
The fixed ratio 1:1:√2 for leg:leg:hypotenuse in a 45-45-90 triangle lets you convert between leg and hypotenuse lengths by multiplying or dividing by √2. If you know the hypotenuse, you can find the leg length by dividing the hypotenuse by √2 (rationalizing the denominator gives hypotenuse × √2 / 2).
2
Label Interpretation in Geometric Diagrams
When solving triangle side length problems, it is critical to match variable labels (x, y) to the correct sides in the diagram. In this problem, x is labeled as the hypotenuse, so it must follow the hypotenuse rule for 45-45-90 triangles.
3
Rationalizing Radicals
When simplifying expressions with radicals in the denominator, rationalize by multiplying numerator and denominator by the radical. For example, if hypotenuse was 8, leg length would be 8/√2 = (8×√2)/(√2×√2) = 8√2/2 = 4√2, which is a simplified radical form.
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