Question

One leg of a right triangle is \(3 \mathrm{cm}\) greater than the other leg, and the hypotenuse is \(15 \mathrm{cm} .\) Find the legs of the triangle.

Short answer

The correct lengths of the two legs are 9 cm and 12 cm.

Step-by-step solution

Assign Variables

Let the length of one leg be x cm. Therefore, the length of the other leg, which is 3 cm longer, will be x+3 cm.

Use the Pythagorean Theorem

Apply the Pythagorean theorem a^2 + b^2 = c^2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Our equation becomes x^2 + (x + 3)^2 = 15^2.

Expand and Simplify the Equation

Expand the equation to x^2 + x^2 + 6x + 9 = 225. Combine like terms to get 2x^2 + 6x + 9 = 225.

Subtract 225 from Both Sides

Subtract 225 from both sides of the equation to get 2x^2 + 6x - 216 = 0.

Factor the Quadratic Equation

Factor the quadratic equation to find the values of x. The factored form will be (2x + 18)(x - 12) = 0.

Solve for x

Set each factor equal to zero and solve for x: 2x + 18 = 0 and x - 12 = 0. The possible values for x are -9 cm and 12 cm. Since a leg length can't be negative, the leg length must be 12 cm.

Calculate the Length of the Other Leg

Now, calculate the length of the other leg which is 3 cm greater than x. Therefore, the other leg is 12 cm + 3 cm = 15 cm. This can't be correct because the other leg can't have the same length as the hypotenuse, discard this solution as an error due to misinterpretation. Correctly, the other leg is 12 cm + 3 cm = 12 cm + 3 cm = 15 cm, the result is incorrect because the other leg cannot be equal to the hypotenuse, a right triangle cannot have two equal sides with one of them being the hypotenuse, there must be an error in the calculation.

Review and Find the Correct Solution

Upon reviewing step 5, we notice that the correct factorization should have led to the equation (2x - 18)(x + 12) = 0. Solve for x to get two possible solutions: x = 9 cm for the first leg, and x = -12 cm which is not possible. Using x = 9 cm, the other leg is then 9 cm + 3 cm = 12 cm, which are the lengths of the two legs.

Key concepts

Right Triangle

Understanding the properties of a right triangle is fundamental in geometry. A right triangle is a type of triangle that has one angle exactly equal to 90 degrees, commonly referred to as the right angle. The two sides that form the right angle are called the 'legs', and the side opposite the right angle is known as the 'hypotenuse', which is always the longest side of the triangle.

In the context of our exercise, we are trying to find the length of the legs when we know that one leg is 3 cm longer than the other, and the hypotenuse is 15 cm long. This setup is a perfect scenario where understanding the properties of right triangles can help us solve for the unknown leg lengths.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, especially when dealing with right triangles. It states that for any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be written as the equation: \( a^2 + b^2 = c^2 \).

By applying this theorem to the problem at hand, we can set up an equation that relates the lengths of the legs and the hypotenuse to find the unknown measurements. It serves as an excellent method for mathematical problem-solving where spatial relationships are involved.

Quadratic Equation

When we encounter an equation where the highest exponent of the variable is 2, we are dealing with a quadratic equation. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where a, b, and c are constants, and x represents an unknown variable.

In the context of the given exercise, by using the Pythagorean theorem, we end up with a quadratic equation that we need to solve to find the lengths of the legs of the right triangle. Quadratic equations can be solved by various methods including factoring, using the quadratic formula, or completing the square.

Factoring Quadratics

Factoring is a useful technique for solving quadratic equations, especially when they can be written in a factorable form. It involves breaking down the equation into a product of two binomials. If we successfully factor the quadratic equation, we can find the solutions by setting each factor to zero and solving for x.

In solving the problem, an error was made during the factoring process. The correct factorization would lead us to the lengths of the legs of the triangle, but care must be taken to ensure the factors are found correctly to avoid incorrect solutions.

Mathematical Problem-Solving

Mathematical problem-solving is a systematic process that often involves understanding the problem, devising a plan, carrying out the plan, and then looking back to check and interpret the solutions. Errors in any of these stages can lead to incorrect answers.

It is important, as seen in our example, to review each step when an unexpected answer is encountered. In the exercise, we expected two positive leg lengths, but initially obtained one leg length equal to the hypotenuse, which does not fit the conditions of a right triangle. This discrepancy provides an opportunity to backtrack and identify any error made. Through a careful review, the correct factors of the quadratic equation are found, leading to the accurate lengths of the triangle's legs.