Problem 66 Suppose that one leg of a right ... [FREE SOLUTION]

91影视

Elementary Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 6: Problem 66

Suppose that one leg of a right triangle is 7 meters longer than the other leg. The hypotenuse is 1 meter longer than the longer leg. Find the lengths of all three sides of the right triangle.

Short Answer

Expert verified

The sides are 5 meters, 12 meters, and 13 meters.

Step by step solution

Set Up Equations

Let's denote the shorter leg of the triangle as $ x $. Therefore, the longer leg will be $ x + 7 $ (since it is 7 meters longer than the other leg). The hypotenuse is 1 meter longer than the longer leg, so it is $ (x + 7) + 1 = x + 8 $.

Apply Pythagorean Theorem

The Pythagorean theorem states that for a right triangle with legs $ a $ and $ b $, and hypotenuse $ c $, the equation $ a^2 + b^2 = c^2 $ holds. In our case:\[ x^2 + (x + 7)^2 = (x + 8)^2 \]

Expand and Simplify the Equation

Expand $ (x + 7)^2 $ and $ (x + 8)^2 $:\[ (x + 7)^2 = x^2 + 14x + 49 \]\[ (x + 8)^2 = x^2 + 16x + 64 \]Substitute these into the equation:\[ x^2 + x^2 + 14x + 49 = x^2 + 16x + 64 \]

Solve for $ x $

Combine like terms and simplify:\[ 2x^2 + 14x + 49 = x^2 + 16x + 64 \]Subtract $ x^2 + 16x + 64 $ from both sides:\[ x^2 - 2x - 15 = 0 \]

Factor the Quadratic Equation

The quadratic equation $ x^2 - 2x - 15 = 0 $ can be factored as:\[ (x - 5)(x + 3) = 0 \]Set each factor to zero: $ x - 5 = 0 $ or $ x + 3 = 0 $. This gives $ x = 5 $ or $ x = -3 $.

Verify the Possible Solution

Since a length cannot be negative, we choose $ x = 5 $. The shorter leg is $ x = 5 $ meters. The longer leg is $ x + 7 = 12 $ meters, and the hypotenuse is $ x + 8 = 13 $ meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Quadratic Equations

Quadratic equations are algebraic expressions characterized by the presence of a squared term as well as linear and constant terms. The typical format of a quadratic equation is:

$ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are constants, and $ a eq 0 $.

To solve a quadratic equation, one can use several methods:

**Factoring**: Expressing the quadratic in the form $(p)(q) = 0$, and solving for $x$.
**Quadratic Formula**: Using the formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ to find solutions.
**Completing the Square**: Rearranging the equation to make one side a perfect square trinomial.

In our context, the quadratic equation $ x^2 - 2x - 15 = 0 $ can be easily solved by factoring. Here, we rewrite it as $(x - 5)(x + 3) = 0$, yielding solutions of $ x = 5 $ or $ x = -3 $. As lengths cannot be negative, $x = 5$ was chosen for this triangle. Understanding how to tackle quadratics expands problem-solving skills in a wide range of geometrical and applied math situations.

Right Triangle

A right triangle is a triangle in which one angle is exactly 90 degrees.

It consists of two legs that meet at the right angle and a hypotenuse, which is the side opposite the right angle and is always the triangle's longest side.

Right triangles can be solved and understood using the Pythagorean Theorem:

Formula: $ a^2 + b^2 = c^2 $, where $ a $ and $ b $ are the legs, and $ c $ is the hypotenuse.

This relationship allows us to determine unknown side lengths given two sides. In our example:

The legs were set as $ x $ and $ x + 7 $.
The hypotenuse was $ x + 8 $.

The theorem demonstrates that the sum of the squares of the legs' lengths equals the square of the hypotenuse's length, guiding us to arrange and solve the quadratic equation.

Geometry Problem

Working with geometry problems often involves identifying relationships between shapes and applying known formulas to find solutions.

For example, understanding characteristics of shapes, such as with a right triangle, involves recognizing key properties and relationships.

In this exercise, we used:

The definition of leg and hypotenuse to define their lengths in terms of known variables.
The Pythagorean Theorem to establish an equation that represents those relationships.

Geometry problems often require combining algebraic skills with spatial reasoning. By integrating these approaches, one can effectively solve for unknowns, whether they're side lengths in triangles or other measurements in different geometrical contexts. Patience and practice with these problems ensure a deeper understanding of mathematical principles.

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91影视

Suppose that one leg of a right triangle is 7 meters longer than the other leg. The hypotenuse is 1 meter longer than the longer leg. Find the lengths of all three sides of the right triangle.

Short Answer

Step by step solution

Set Up Equations

Apply Pythagorean Theorem

Expand and Simplify the Equation

Solve for \( x \)

Factor the Quadratic Equation

Verify the Possible Solution

Key Concepts

Solving Quadratic Equations

Right Triangle

Geometry Problem

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