The Length of the Shorter Leg in a Right Triangle with a Given Ratio

When dealing with right triangles where the legs are in a known ratio, it is often necessary to find the exact lengths of the legs, especially when the hypotenuse is also provided. This article will guide you through the process using an example involving the legs of a right triangle in the ratio 5:12 and a hypotenuse of 65 feet.

Understanding the Problem

The problem states that the legs of a right triangle are in the ratio 5:12, and the hypotenuse is 65 feet. The goal is to find the length of the shorter leg.

Step-by-Step Solution

Define the Legs in Terms of a Variable

Let the shorter leg be (5x) and the longer leg be (12x). This step allows us to use algebra to set up the equation needed to solve for (x).

Apply the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle with legs (a) and (b), and hypotenuse (c), the relationship is given by:

[a^2 b^2 c^2]

Here, we have:

[5x^2 12x^2 65^2]

After substituting the values, the equation becomes:

Calculate the Squares

[ (5x)^2 (12x)^2 65^2 ]

[ 25x^2 144x^2 4225 ]

[ 169x^2 4225 ]

Solve for (x^2)

[ x^2 frac{4225}{169} ]

[ x^2 25 ]

[ x 5 ]

Calculate the Lengths of the Legs

Now that we have (x 5), we can calculate the lengths of the legs:

[ text{Shorter leg} 5x 5 times 5 25 text{ ft} ]

[ text{Longer leg} 12x 12 times 5 60 text{ ft} ]

Therefore, the length of the shorter leg is 25 feet.

Conclusion

By applying the Pythagorean theorem and solving the resulting equation, we were able to determine the length of the shorter leg of the right triangle given its ratio and the hypotenuse. This process provides a clear method for solving similar problems involving right triangles with known ratios.

Additional Tips

Knowing common Pythagorean triplets (such as 5:12:13) can make solving these problems more straightforward. If the legs of the triangle are in the ratio 5:12, and the hypotenuse is 13 times the smallest leg, then the smallest leg must be the same as in the given problem.

For the given problem, the 5:12:13 triplet fits the description, so the smaller leg would be 5 times the smallest leg in the triplet, which is 5 ft, giving the length of the shorter leg as 25 feet.