Problem 79 - Algebra and Trigonometry

Question

Dimensions of a Lot \(\quad\) A city lot has the shape of a right triangle whose hypotenuse is 7 ft longer than one of the other sides. The perimeter of the lot is 392 ft. How long is each side of the lot?

Step-by-Step Solution

Answer

The sides of the lot are 119 ft, 147 ft, and 126 ft.

Step 1: Understand the Problem

We need to find the lengths of the sides of a right triangle where the hypotenuse is 7 feet longer than one of the other sides, and the perimeter is 392 feet.

Step 2: Define Variables

Let's denote one of the sides as \(a\), the other side as \(b\), and the hypotenuse as \(c\). According to the problem, \(c = a + 7\).

Step 3: Establish the Perimeter Equation

The perimeter of the triangle is the sum of all its sides: \(a + b + c = 392\). Since \(c = a + 7\), the equation becomes \(a + b + (a + 7) = 392\).

Step 4: Simplify the Perimeter Equation

Simplify the equation from the previous step: \(2a + b + 7 = 392\). Now, simplify it further to \(2a + b = 385\).

Step 5: Use Pythagoras' Theorem

Since we have a right triangle, we can use Pythagoras' theorem: \(a^2 + b^2 = c^2\). Substitute \(c = a + 7\) into the equation to get \(a^2 + b^2 = (a + 7)^2\).

Step 6: Expand Pythagorean Equation

Expand \((a + 7)^2\) to get \(a^2 + b^2 = a^2 + 14a + 49\).

Step 7: Solve for \(b\)

Subtract \(a^2\) from both sides of the equation: \(b^2 = 14a + 49\). Use the equation from Step 4: \(2a + b = 385\) to express \(b\) in terms of \(a\): \(b = 385 - 2a\).

Step 8: Substitute \(b\) into the Pythagorean Equation

Substitute \(b = 385 - 2a\) into \(b^2 = 14a + 49\): \((385 - 2a)^2 = 14a + 49\). Expand and solve this quadratic equation for \(a\).

Step 9: Solve the Quadratic Equation

After expansion, solve the quadratic equation: \(4a^2 - 1540a + 148225 = 14a + 49\).Simplify to get: \(4a^2 - 1554a + 148176 = 0\).Use the quadratic formula, where \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to solve for \(a\).

Step 10: Calculate and Verify Sides

Solve the quadratic equation and find \(a\), then calculate \(b = 385 - 2a\) and \(c = a + 7\). Verify by adding \(a\), \(b\), and \(c\) to ensure that the sum is 392.

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is a key concept in geometry. It is simply the total distance around the three sides of the triangle. To find the perimeter, you add up the lengths of all three sides. In a right triangle problem, understanding the perimeter is crucial because it lets you set up equations that help find unknown side lengths.

For instance, in our problem, the perimeter of the right triangle is 392 feet. This gives the equation:

• Add the lengths of all three sides together: \(a + b + c = 392\).
• By understanding this equation, you can substitute known expressions to find unknown quantities, which is central to geometry problem-solving.

Pythagorean Theorem

The Pythagorean theorem is a powerful tool for solving geometry problems involving right triangles. It states that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This is expressed as:

• \( a^2 + b^2 = c^2 \)

This theorem allows you to solve for one side of a triangle when you know the lengths of the other two. In this problem, we need to apply the theorem because it involves checking the relationship between the sides of the right triangle.

Understanding this principle and being able to apply it correctly is essential for anyone studying geometry, as it forms the foundation for working with right triangles.

Quadratic Equations

Quadratic equations appear frequently in problems involving geometry, especially right triangle problems. A quadratic equation is any equation that can be rearranged in the standard form:

• \( ax^2 + bx + c = 0 \)

To solve these equations, you can either factor them, complete the square, or use the quadratic formula. The quadratic formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In this exercise, after setting up the equations involving the triangle's sides, we arrive at a quadratic equation to solve for one of the sides. The ability to solve quadratic equations is crucial, as it often leads you to the solution of the problem when dealing with polynomial expressions in geometry.

Geometry Problem-Solving

Solving geometry problems often involves combining different concepts and tools like the perimeter, Pythagorean theorem, and quadratic equations. It's important to systematically approach the problem by breaking it into smaller, manageable parts.

Here are some steps that can help improve your geometry problem-solving skills:

• First, understand what the problem is asking by identifying all known and unknown quantities.
• Next, choose suitable mathematical tools such as the Pythagorean theorem or perimeter formulas to set up equations.
• Then, simplify these equations and solve them using appropriate methods like factoring or the quadratic formula.
• Finally, verify your solution by checking that it satisfies all conditions given in the problem.

Developing these strategies not only helps with specific problems but also strengthens overall mathematical reasoning and logic.