Question

Find the missing side and the hypotenuse of a right triangle that has one side equal to 293 in. and a perimeter of 994 in.

Short answer

Solve the equation b + c = 701 for c, plug the value back into the Pythagorean theorem to find that side b is approximately 504 inches and the hypotenuse c is about 197 inches. The triangle's sides are 293 in, 504 in, and 197 in.

Step-by-step solution

Understand the properties of a right triangle

A right triangle consists of two legs and a hypotenuse, which is the longest side. The perimeter (P) of the triangle is the sum of the lengths of all three sides. If the length of one leg is given as 293 inches, we can represent the unknown leg as 'b' and the hypotenuse as 'c'. The perimeter equation is therefore represented as P = 293 + b + c.

Set up the equation with the given perimeter

With the perimeter given as 994 inches, we can set up the equation as 994 = 293 + b + c and simplify it to find a relationship between b and c, which gives us b + c = 994 - 293.

Use the Pythagorean Theorem

The Pythagorean theorem states that in a 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. This can be written as c^2 = 293^2 + b^2.

Express the hypotenuse in terms of the other unknown

From Step 2, we can express c as c = 701 - b. Substituting this expression for c into the Pythagorean theorem from Step 3, we obtain (701 - b)^2 = 293^2 + b^2.

Solve for the unknown side 'b'

Expand and simplify the equation from Step 4 to solve for b. Doing the algebra will eventually get you a quadratic equation in b, which you can solve through factorization or using the quadratic formula.

Substitute the value of 'b' to find the hypotenuse

Once you've obtained the value of b, substitute it back into the expression for c from Step 4 to find the length of the hypotenuse.

Double-check the solution

Finally, ensure that the sum of the sides equals the perimeter and that the Pythagorean theorem holds true. These checks will validate that the values found for b and c are correct.

Key concepts

Right Triangle Properties

Understanding the properties of a right triangle is fundamental when solving related geometry problems. Firstly, a right triangle consists of two shorter sides known as the legs and the longest side, called the hypotenuse. The hypotenuse lies opposite the triangle's right angle. Importantly, the other two angles in a right triangle are always acute, which means they are less than 90 degrees.

In the context of the given problem, one leg length is known, and the relationship between the other leg (denoted as 'b'), the hypotenuse ('c'), and the perimeter (which is the sum of all three sides) is essential for finding the missing measurements.

Pythagorean Theorem

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

This theorem is applied directly to the problem at hand, enabling the establishment of a relationship between the given leg (293 inches) and the unknown leg and hypotenuse. Once these relationships have been detailed, the theorem becomes a pivotal tool in calculating the missing side lengths.

Quadratic Equation

When two variables are involved in a problem that result in an equation where the unknown is squared, you're often looking at 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 the unknown variable.

In the right triangle perimeter problem, applying the Pythagorean theorem eventually leads to a quadratic equation in terms of 'b', the unknown leg. This equation will often need to be solved using factorization or the quadratic formula, which is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). It’s essential to handle the algebra with care to ensure solving for 'b' correctly.

Triangle Perimeter Calculation

The perimeter of a triangle is the total distance around the triangle, which is the sum of all its sides. For the problem given, the perimeter is explicitly stated, allowing us to set up an equation to find the unknown sides. The perimeter calculation in a right triangle involves adding the two legs and the hypotenuse.

In the step-by-step solution, once the value of 'b' is obtained, it needs to be substituted back to get the measurement of the hypotenuse. It's this addition of the known side, the found leg 'b', and the hypotenuse 'c' that should equate to the given perimeter, thus ensuring the solution is correct and adheres to the properties of a right triangle.