Problem 6 - Prealgebra

Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. $$5 \mathrm{cm}, 7 \mathrm{cm}, 8 \mathrm{cm}$$

Step-by-Step Solution

Answer

The triangle with sides 5 cm, 7 cm, and 8 cm is not a right triangle.

Step 1: Identify the Triangle Sides

The given lengths are 5 cm, 7 cm, and 8 cm. We need to determine which one is the hypotenuse by identifying the largest side, which in this case is 8 cm.

Step 2: Recall the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides: \( c^2 = a^2 + b^2 \). In this case, let \( c = 8 \), \( a = 5 \), and \( b = 7 \).

Step 3: Calculate the Squares of Each Side

Calculate the square of each side:- \( 8^2 = 64 \) (hypotenuse)- \( 5^2 = 25 \)- \( 7^2 = 49 \)

Step 4: Apply the Pythagorean Theorem

Check if the sum of the squares of the two shorter sides equals the square of the hypotenuse:\( 5^2 + 7^2 = 25 + 49 = 74 \).Compare that to \( 8^2 = 64 \). Since 74 is not equal to 64, the triangle is not a right triangle.

Key Concepts

Understanding the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially when it comes to right triangles. Essentially, it states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This relationship is expressed mathematically as \( c^2 = a^2 + b^2 \), where:

• \( c \) represents the hypotenuse.
• \( a \) and \( b \) are the other two sides of the triangle.

To determine whether a given triangle is a right triangle, you can plug the lengths of the sides into this equation. If the equality holds true, then the triangle is a right triangle. Otherwise, it is not. The theorem only applies to right triangles, making it a useful tool in verifying if a triangle with known side lengths forms a right angle.

Measuring Triangle Sides

Every triangle consists of three sides, and in the context of the Pythagorean theorem, we need to identify which side is the hypotenuse. Generally, the hypotenuse is the longest side.
In our exercise, the provided side lengths were 5 cm, 7 cm, and 8 cm, making 8 cm the hypotenuse since it is the largest value. In a practical situation, measuring triangle sides accurately is vital to apply the theorem correctly.

• Use measuring tools such as a ruler or tape measure to get precise side lengths.
• Ensure that the triangle is properly formed to avoid errors in identification.

Accurate measurement ensures the correct application of the Pythagorean theorem, aiding in determining whether a given triangle is right or not.

Determining the Hypotenuse

In geometry, the term hypotenuse refers to the longest side of a right triangle. When presented with three side lengths, identifying the hypotenuse is the first step in applying the Pythagorean theorem. If the triangle is a right triangle, the hypotenuse acts as the basis for calculating the sum of the squares of the other two sides, which should equal its square.For example, using the side lengths 5 cm, 7 cm, and 8 cm:

• Hypotenuse \( = 8^2 \)
• Other two sides are \( 5^2 \) and \( 7^2 \)

Checking if the square of 8 equals the sum of the squares of 5 and 7 helps determine if the triangle is right. This step is crucial as it either confirms or refutes the presence of a right angle, based on whether the calculated values match up.