Question

What is the perimeter of a right triangle whose height is twice its base and whose area is 72.25 square inches? A. 8.5 inches B. 17 inches C. 19 inches D. 44.5 inches

Short answer

44.5 inches.

Step-by-step solution

- Define the Variables

Let the base of the right triangle be denoted as \( b \) and the height as \( h \). According to the problem, the height \( h \) is twice the base \( b \). Therefore, \( h = 2b \).

- Use the Area Formula

The area of the right triangle is given by the formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Plugging in the known values, we get: \( 72.25 = \frac{1}{2} \times b \times 2b \).

- Solve for the Base

Simplify the equation \( 72.25 = b^2 \). To find \( b \), take the square root of both sides: \( b = \sqrt{72.25} = 8.5 \) inches.

- Calculate the Height

Since the height is twice the base, substitute \( b \) into the equation \( h = 2b \): \( h = 2 \times 8.5 = 17 \) inches.

- Apply the Pythagorean Theorem

In a right triangle, the Pythagorean theorem states \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Use this to find the hypotenuse \( c \): \( 8.5^2 + 17^2 = c^2 \). Calculate each term: \( 72.25 + 289 = c^2 \). So, \( c^2 = 361.25 \). Taking the square root gives: \( c = \sqrt{361.25} = 19 \) inches.

- Find the Perimeter

The perimeter of a triangle is the sum of all its sides: \( \text{Perimeter} = b + h + c \). Substitute the values: \( 8.5 + 17 + 19 = 44.5 \) inches.

Key concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry, especially for right triangles. It relates the lengths of the sides of a right triangle. If you know two sides, you can find the third one using this theorem. The theorem is given by the equation: \(a^2 + b^2 = c^2\)where

• \(a\) and \(b\) are the legs (the two shorter sides) of the triangle
• \(c\) is the hypotenuse (the longest side opposite the right angle)

For example, in the given exercise, the legs are 8.5 inches and 17 inches. Plugging these into the Pythagorean theorem, we get:\[8.5^2 + 17^2 = c^2\]This simplifies to:\[72.25 + 289 = c^2\]Thus:\(c^2 = 361.25\)and the hypotenuse \(c = \sqrt{361.25} = 19\) inches.

triangle area formula

The area of a triangle can be calculated using the triangle area formula. For right triangles, this formula is particularly simple. It is given by:\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)If you know the area and one dimension (base or height), you can find the other dimension by rearranging the formula. In our exercise, the area is 72.25 square inches and the height is twice the base. Let the base be \(b\) and the height be \(h\); we have:\[72.25 = \frac{1}{2} \times b \times 2b\]Simplifying, we find:\[72.25 = b^2\]Solving for \(b\), we get:\[b = \sqrt{72.25} = 8.5\]The height \(h\) is twice the base, so:\[h = 2 \times 8.5 = 17\] inches.

solving quadratic equations

Quadratic equations are equations of the form:\(ax^2 + bx + c = 0\)They can be solved using various methods including factoring, completing the square, or the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In the exercise, we encounter a simplified quadratic equation,\(b^2 = 72.25\).This is already in a form where we can directly solve for \(b\) by taking the square root of both sides:\[b = \sqrt{72.25}\]which simplifies to:\[b = 8.5\] inches.Quadratic equations appear in many areas of mathematics and physics and knowing how to solve them is a crucial skill.