Chapter 13: Problem 4
The area of a right triangle is 35 square inches. If the longer leg is 3 inches longer than the shorter leg, what is the length of the hypotenuse, in inches? A. 10 B. \(\frac{\sqrt{35}}{2}\) C. \(7 \sqrt{10}\) D. \(\sqrt{149}\)
Short Answer
Expert verified
D. \( \root 149 \)
Step by step solution
01
Identify Variables
Let the shorter leg be denoted as x inches. Consequently, the longer leg will be x + 3 inches.
02
Recall Area Formula for Right Triangle
The area of a right triangle is given by \( \frac{1}{2} \times \text{base} \times \text{height} \). In this case, the base and height are the legs of the triangle.
03
Set Up Equation for Area
According to the problem, the area is 35 square inches. Therefore, we set up the equation: \[ \frac{1}{2} \times x \times (x + 3) = 35 \]
04
Solve for x
First, multiply both sides by 2 to clear the fraction: \[ x(x + 3) = 70 \]. Next, distribute and set the equation to zero: \[ x^2 + 3x - 70 = 0 \]. Solve the quadratic equation \( x^2 + 3x - 70 = 0 \) using the quadratic formula \( x = \frac{-b \, \text{±}\, \root b^2 - 4ac}{2a} \), where a = 1, b = 3, and c = -70.
05
Calculate Roots
Using the quadratic formula, find x. \( x = \frac{-3 \, \text{±}\, \root 9 + 280}{2} \), which simplifies to \( x = \frac{-3 \, \text{±}\, \root 289}{2} \). This results in \( x = \frac{-3 \, \text{±}\, 17}{2} \). Thus, \( x = 7 \) or \( x = -10 \). Since a leg length cannot be negative, \( x = 7 \).
06
Determine Leg Lengths
The shorter leg is 7 inches, and the longer leg is \( 7 + 3 = 10 \) inches.
07
Use Pythagorean Theorem
To find the hypotenuse, use the Pythagorean theorem: \( a^2 + b^2 = c^2 \). Here, \( a = 7 \) and \( b = 10 \). Thus, \( 7^2 + 10^2 = c^2 \) or \( 49 + 100 = c^2 \). Therefore, \( c^2 = 149 \) and \( c = \root 149 \).
08
Identify Correct Answer
The length of the hypotenuse is \( \root 149 \), which corresponds to option D.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Right Triangle Area
The area of a right triangle is found using the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] This formula works because a right triangle is half of a rectangle whose area is base times height. In this problem, the base and height are the legs of the triangle. Knowing one leg is 3 inches longer than the other helps us set up the equation. This is a key concept in finding the variables needed to solve the problem.
Quadratic Equation
Quadratic equations are fundamental in math, especially for problems like this. A quadratic equation is in the form: \[ ax^2 + bx + c = 0 \] Solving quadratic equations often involves the quadratic formula: \[ x = \frac{-b \, ±\, \sqrt{b^2 - 4ac}}{2a} \] For this exercise, the equation is set up from the triangle's area formula: \[ x(x + 3) = 70 \] This simplifies to: \[ x^2 + 3x - 70 = 0 \] Using the quadratic formula, we solve for x: \[ x = \frac{-3 \, ±\, \sqrt{9 + 280}}{2} = \frac{-3 \, ±\, 17}{2} \] Choosing the positive root, we find x = 7, which is the shorter leg.
Pythagorean Theorem
The Pythagorean theorem is crucial for finding the hypotenuse in a right triangle. It states: \[ a^2 + b^2 = c^2 \] where a and b are the legs, and c is the hypotenuse. With the leg lengths found (7 and 10 inches), we apply the theorem: \[ 7^2 + 10^2 = c^2 \] So: \[ 49 + 100 = c^2 \] Therefore: \[ c^2 = 149 \] Finally, the hypotenuse c is: \[ c = \sqrt{149} \] This theorem is a cornerstone for solving right triangle problems on tests like the SAT.
SAT Prep
Preparing for the SAT involves mastering key math concepts, including right triangle area, quadratic equations, and the Pythagorean theorem. To effectively study:
- Practice solving for variable values in area and geometry problems.
- Familiarize yourself with the quadratic formula and solving quadratic equations.
- Understand and apply the Pythagorean theorem in various problems.