Math Problem Statement

Give an expression for the length of the missing hypotenuse.

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:

  1. Identify the smaller triangle:

    • The smaller triangle is a 45-45-90 triangle with legs of length 1 and hypotenuse of 2\sqrt{2}.
  2. Use the properties of similar triangles:

    • In a 45-45-90 triangle, the legs are equal, and the hypotenuse is 2\sqrt{2} times the length of each leg.
    • The larger triangle is also a 45-45-90 triangle since its legs are both equal (6).
  3. Set up the proportion:

    • The ratio of the side lengths in any 45-45-90 triangle is 1:1:21 : 1 : \sqrt{2}.
    • Since each leg of the larger triangle is 6, the hypotenuse of the larger triangle will be 6×26 \times \sqrt{2}.

Conclusion:

The length of the missing hypotenuse is: 626\sqrt{2}

Would you like more details, or do you have any questions?


Related Questions:

  1. How would you find the hypotenuse of a right triangle if only one leg is given?
  2. What is the relationship between the sides of a 30-60-90 triangle?
  3. How can similarity of triangles be used to solve for unknown lengths?
  4. Why is the ratio 1:1:21 : 1 : \sqrt{2} specific to 45-45-90 triangles?
  5. 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 2\sqrt{2}.

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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