Question

A right triangle's hypotenuse is 10 inches long. The length of one leg of a right triangle is 2 inches more than the second leg. What are the lengths of the legs of the triangle?

Answer 1

Answer:

  8 inches and 6 inches

Step-by-step explanation:

The hypotenuse length of 10 is 2 times 5, so there is a possibility that the triangle is a 3-4-5 triangle scaled by a factor of 2. The leg length difference of 2 confirms this. The right triangle of interest has sides of 6 inches, 8 inches, and 10 inches, double the sides of a 3-4-5 triangle.

Since "one leg" is 2 inches longer, it must be the 8-inch leg. The "second leg" must be the 6-inch leg.

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The 3-4-5 right triangle shows up in many algebra and geometry problems, so is useful to remember.

If you want to solve this algebraically, you can assign x as the length of "one leg" and (x -2) as the length of "the second leg." Then the Pythagorean theorem tells you ...

  10² = x² +(x -2)²

  100 = 2x² -4x +4

  x² -2x -48 = 0 . . . . . subtract 100, divide by 2

  (x -8)(x +6) = 0 . . . . .factor

  x = 8 . . . . . . . . . . . . . the positive solution; the only one of interest

One leg is 8 inches; the second leg is 6 inches.

Answer 2

6 inches and 8 inches.

Further explanation

Given:

• A right triangle's hypotenuse is 10 inches long.
• The length of one leg of a right triangle is 2 inches more than the second leg.

Question:

What are the lengths of the legs of the triangle?

The Process:

We will solve a problem regarding right triangles. In the process, there is a relationship between the Pythagorean Theorem with Quadratic Equations.

Let x as the length of one leg of a right triangle. Thus, the length of the second leg is (x - 2).

All length units were expressed in inches.

Let us find out the lengths of the legs of the triangle.

The Phytagorean Theorem: $\boxed{ \ (leg)^2 + (leg)^2 = (hypotenuse)^2 \ }$

$\boxed{ \ (x-2)^2 + (x)^2 = (10)^2 \ }$

Remember this, $\boxed{\boxed{ \ (a - b)^2 = a^2 - 2ab + b^2\ } \ \boxed{ \ (a +b)^2 = a^2 + 2ab + b^2\ }}$

$\boxed{ \ x^2 - 4x + 4 + x^2 = 100 \ }$

$\boxed{ \ 2x^2 - 4x + 4 - 100= 0 \ }$

$\boxed{ \ 2x^2 - 4x - 96= 0 \ }$

Divide by two on both sides.

$\boxed{ \ x^2 - 2x - 48= 0 \ }$

$\boxed{ \ (x - 8)(x + 6) = 0 \ }$

• The accepted root represents x = 8 inches long because the length must be positive.
• Thus, the length of the second leg is 8 - 2 = 6 inches long.

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Notes

Of course, we agree with the use of the right intuition, which is to remember Pythagorean Triples, i.e., $\boxed{ \ 3-4-5 \ } \rightarrow \boxed{ \ 6-8-10 \ }$ This method is very convenient when working on multiple-choice tests in a limited amount of time.

Learn more

1. Find out the measures of the two angles in a right triangle brainly.com/question/4302397
2. A word problem of a quadratic equation brainly.com/question/11805547
3. What is the area of triangle ABC ? brainly.com/question/4206319