Question

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

D. \( \root 149 \)

Step-by-step solution

Identify Variables

Let the shorter leg be denoted as x inches. Consequently, the longer leg will be x + 3 inches.

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.

Set Up Equation for Area

According to the problem, the area is 35 square inches. Therefore, we set up the equation: \[ \frac{1}{2} \times x \times (x + 3) = 35 \]

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.

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, \( x = 7 \).

Determine Leg Lengths

The shorter leg is 7 inches, and the longer leg is \( 7 + 3 = 10 \) inches.

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 \( c = \root 149 \).

Identify Correct Answer

The length of the hypotenuse is \( \root 149 \), which corresponds to option D.

Key concepts

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.

Regular practice with these concepts will boost problem-solving speed and confidence, ensuring better performance on the math section of the SAT.