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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A right triangle has a hypotenuse of length 13 inches13\text{ inches}. One of the legs is 7 inches7\text{ inches} longer than the other leg. What is the length, in inches, of the shorter leg?

  1. A
    3.0
  2. 5.0Answer
  3. C
    6.0
  4. D
    8.6
  5. E
    11.0

Answer

The length of the shorter leg is 5.0 inches.
The correct answer is the option representing 5.0. By setting the shorter leg as xx, the longer leg is x+7x + 7. Applying the Pythagorean theorem gives x2+(x+7)2=132x^2 + (x + 7)^2 = 13^2, which simplifies to 2x2+14x−120=02x^2 + 14x - 120 = 0. Factoring the divided equation x2+7x−60=0x^2 + 7x - 60 = 0 gives (x−5)(x+12)=0(x - 5)(x + 12) = 0. Since length must be positive, the shorter leg is 5.0 inches.

Step-by-Step Solution

1
Define variables for the side lengths of the right triangle based on the problem statement.
Let the length of the shorter leg be xx inches. The length of the longer leg is x+7x + 7 inches, and the hypotenuse is 1313 inches.
This translates the verbal descriptions into mathematical expressions using a single variable.
2
Apply the Pythagorean theorem to set up an equation relating the side lengths.
x2+(x+7)2=132x^2 + (x + 7)^2 = 13^2
For any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.
3
Expand the squared binomial and simplify the equation into standard quadratic form.
x2+(x2+14x+49)=169⇒2x2+14x−120=0x^2 + (x^2 + 14x + 49) = 169 \Rightarrow 2x^2 + 14x - 120 = 0
Expanding allows us to group like terms and solve for the variable xx.
4
Divide the quadratic equation by 2 and factor the resulting expression.
x2+7x−60=0⇒(x+12)(x−5)=0x^2 + 7x - 60 = 0 \Rightarrow (x + 12)(x - 5) = 0
Factoring is the most efficient way to find the roots of this quadratic equation.
5
Solve for xx and select the mathematically and physically valid solution.
x=5x = 5 or x=−12x = -12. Since a side length must be positive, x=5x = 5 inches.
Lengths in geometry must be positive, so we discard the negative root.

Key Concept

Applying the Pythagorean theorem to solve for unknown side lengths of a right triangle given algebraic relationships between the sides.
Estimated Time:1m 30s
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Topics

Pythagorean Theorem and Special Right Triangles

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