Math Problem Statement
Solution
To find the length of the missing hypotenuse in the larger triangle, we can observe that both triangles shown are similar right triangles.
Steps:
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Identify the smaller triangle:
- The smaller triangle is a 45-45-90 triangle with legs of length 1 and hypotenuse of .
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Use the properties of similar triangles:
- In a 45-45-90 triangle, the legs are equal, and the hypotenuse is times the length of each leg.
- The larger triangle is also a 45-45-90 triangle since its legs are both equal (6).
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Set up the proportion:
- The ratio of the side lengths in any 45-45-90 triangle is .
- Since each leg of the larger triangle is 6, the hypotenuse of the larger triangle will be .
Conclusion:
The length of the missing hypotenuse is:
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Related Questions:
- How would you find the hypotenuse of a right triangle if only one leg is given?
- What is the relationship between the sides of a 30-60-90 triangle?
- How can similarity of triangles be used to solve for unknown lengths?
- Why is the ratio specific to 45-45-90 triangles?
- How would you calculate the hypotenuse if the legs had different lengths?
Tip:
In any 45-45-90 triangle, if you know the length of one leg, you can immediately determine the hypotenuse by multiplying the leg length by .
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Math Problem Analysis
Mathematical Concepts
Geometry
Right Triangles
Similarity of Triangles
45-45-90 Triangle
Formulas
Hypotenuse = leg * √2 for 45-45-90 triangles
Theorems
Similarity of Triangles Theorem
Properties of 45-45-90 Triangles
Suitable Grade Level
Grades 8-10
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