Problem 36 - College Algebra with Modeling and Visualization

Question

Use the Pythagorean theorem to find the missing side of the right triangle with legs a and \(b\) and hypotenuse \(c .\) Then calculate the perimeter. Approximate values to the nearest tenth when appropriate. Shorter leg 40 centimeters and hypotenuse twice the shorter leg

Step-by-Step Solution

Answer

The perimeter of the triangle is 189.3 cm.

Step 1: Define the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of its legs (a and b). It is expressed as: \( a^2 + b^2 = c^2 \).

Step 2: Identify given values

We are given the shorter leg \( a = 40 \) cm, and the hypotenuse \( c = 2a = 80 \) cm. We need to find the length of the other leg \( b \).

Step 3: Substitute values in the Pythagorean Theorem

Substitute the known values into the equation: \( 40^2 + b^2 = 80^2 \).

Step 4: Simplify and solve for b

Calculate \(40^2 = 1600\) and \(80^2 = 6400\). Replace these values back into the equation: \( 1600 + b^2 = 6400 \). Subtract 1600 from both sides to isolate \( b^2 \): \( b^2 = 6400 - 1600 \). So, \( b^2 = 4800 \).

Step 5: Find the length of b

Take the square root of both sides to solve for \( b \): \( b = \sqrt{4800} \). Approximating \( \sqrt{4800} \) gives \( b \approx 69.3 \) cm.

Step 6: Calculate the perimeter

The perimeter of the triangle is the sum of all its sides: \( P = a + b + c \). Substitute the known values: \( P = 40 + 69.3 + 80 \). Calculate the perimeter: \( P = 189.3 \) cm.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This special triangle allows for the use of the Pythagorean Theorem to calculate the unknown lengths of its sides. In a right triangle, there are two shorter sides called 'legs' and the longest side is known as the 'hypotenuse.' The angle opposite the hypotenuse is the right angle, making this triangle unique.

When you know the length of two sides of a right triangle, you can use the Pythagorean theorem to find the third side's length. This feature makes right triangles particularly helpful in various fields such as architecture, engineering, and art. Here are a few things to remember:

• The sum of the angles in any triangle is always 180 degrees, so the two non-right angles in a right triangle are complementary (they add up to 90 degrees).
• Right triangles can be scalene, with all sides of different lengths, or isosceles, with two sides of equal length.

Understanding these basics helps in solving problems related to right triangles and applying the Pythagorean theorem effectively.

Hypotenuse

The hypotenuse is the longest side of a right triangle. It is opposite the right angle and plays a crucial role in the Pythagorean theorem. This theorem is formulated as: \[ a^2 + b^2 = c^2 \]where \(a\) and \(b\) are the legs of the triangle and \(c\) is the hypotenuse.

Calculating the hypotenuse involves solving for \(c\) when the lengths of \(a\) and \(b\) are known. In our problem, the hypotenuse \(c\) is given as twice the shorter leg, making it 80 cm. Let's review some points about the hypotenuse in right triangles:

• The hypotenuse is always the longest side in a right triangle.
• The square of the hypotenuse is equal to the sum of the squares of the other two sides, making calculations straightforward with the theorem.
• Applications often involve determining height or distance, using the hypotenuse as the diagonal across those dimensions.

Understanding the role of the hypotenuse makes solving geometric problems using right triangles much more approachable.

Perimeter Calculation

Calculating the perimeter of a triangle involves simply adding up the lengths of all three sides. The perimeter provides the total distance around the triangle, a useful measure in various applications, such as fencing a triangular garden or framing a triangular section in construction.

In the given problem, once we find all the side lengths, calculating the perimeter becomes straightforward. Here's a simple guide:

• First, find or calculate the lengths of all three sides using given measures and equations like the Pythagorean Theorem.
• Sum these lengths: \( a + b + c \). For our exercise, that equates to
  \( P = 40 + 69.3 + 80 = 189.3 \) cm.
• The calculated perimeter gives a real-world sense of how much space the triangle occupies when outlined.

With these steps, understanding and calculating the perimeter of any triangle becomes simpler and helps you see the relationship between geometry and practical applications.