Problem 33 - Beginning and Intermediate Algebra

Question

Solve each problem. When appropriate, round answers to the nearest tenth. A game board is in the shape of a right triangle. The hypotenuse is 2 in. longer than the longer leg, and the longer leg is 1 in. less than twice as long as the shorter leg. How long is each side of the game board?

Step-by-Step Solution

Answer

The sides of the game board are 8 inches, 15 inches, and 17 inches.

Step 1 - Define Variables

Let the shorter leg be denoted as x. The longer leg, being 1 inch less than twice the shorter leg, can be represented as 2x - 1. The hypotenuse is 2 inches longer than the longer leg, so it is represented as (2x - 1) + 2 or 2x + 1.

Step 2 - Use the Pythagorean Theorem

For a right triangle, the Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs: \[ (2x + 1)^2 = x^2 + (2x - 1)^2 \]

Step 3 - Simplify the Equation

Expand and simplify both sides of the equation: \[ (2x + 1)^2 = x^2 + (2x - 1)^2 \] Expand: \[ 4x^2 + 4x + 1 = x^2 + 4x^2 - 4x + 1 \] Combine like terms: \[ 4x^2 + 4x + 1 = 5x^2 - 4x + 1 \]

Step 4 - Solve for x

Subtract 4x^2 + 4x + 1 from both sides to collect all terms on one side: \[ 0 = x^2 - 8x \] Factor the quadratic equation: \[ x(x - 8) = 0 \] Solve for x: \[ x = 0 \text{ or } x = 8 \] Since a side length cannot be zero, we take x = 8.

Step 5 - Determine the Side Lengths

With x = 8, the shorter leg: x = 8 inches. The longer leg: 2x - 1 = 2(8) - 1 = 15 inches. The hypotenuse: 2x + 1 = 2(8) + 1 = 17 inches.

Step 6 - Round if Necessary

All sides are already in whole numbers, so no rounding is needed.

Key Concepts

right triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. It consists of three sides:

• The hypotenuse, which is the longest side and lies opposite the right angle.
• The legs, which are the two shorter sides that form the right angle.

The Pythagorean Theorem is a key concept used to relate the lengths of the sides. In this problem, we denote the shorter leg as x, the longer leg as 2x - 1, and the hypotenuse as 2x + 1. By forming these expressions, we can solve for x and subsequently find the lengths of all sides.

solving quadratic equations

Quadratic equations are equations of the second degree, generally in the form ax^2 + bx + c = 0. Solving them often involves factoring, using the quadratic formula, or completing the square. In this problem, we derive a quadratic equation from the Pythagorean Theorem:

• Expand and combine like terms: \[ (2x + 1)^2 = x^2 + (2x - 1)^2 \rightarrow 4x^2 + 4x + 1 = x^2 + 4x^2 - 4x + 1 \rightarrow 4x^2 + 4x + 1 = 5x^2 - 4x + 1 \]
• Simplify and rearrange terms: \[ 4x^2 + 4x + 1 - (4x^2 + 4x + 1) = 5x^2 - 4x + 1 - (4x^2 + 4x + 1) \rightarrow 0 = x^2 - 8x \]

Finally, we factor and solve the quadratic equation to find x.

algebraic expressions

Algebraic expressions help to represent quantities and their relationships using variables and constants. In the exercise, we use algebraic expressions to define the lengths of the sides of the right triangle:

• The shorter leg is represented by x.
• The longer leg is 1 inch less than twice the shorter leg, represented as 2x - 1.
• The hypotenuse is 2 inches longer than the longer leg, represented as 2x + 1.

These expressions are then substituted into the Pythagorean Theorem equation to form an algebraic equation that we solve to determine the lengths of the sides.

geometry

Geometry is a branch of mathematics concerned with shapes, sizes, and the properties of space. Right triangles are fundamental objects in geometry due to their unique properties and importance in various applications. This exercise demonstrates how geometric principles, specifically the Pythagorean Theorem, can be used to solve for unknown side lengths when certain conditions are known. By understanding the relationships between the sides of a right triangle, we can apply algebraic methods to find the exact dimensions of the shape:

• Understand how the Pythagorean Theorem relates to a right triangle's sides.
• Use algebraic manipulation to solve for unknowns given specific geometric properties.

Geometry thus blends spatial reasoning with algebraic techniques to solve complex problems effectively.