Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. \(6 \mathrm{mi}, 7 \mathrm{mi}, 12 \mathrm{mi}\)

Short answer

The triangle is not a right triangle.

Step-by-step solution

Identify the sides

For the sides of the triangle given as 6 mi, 7 mi, and 12 mi, we need to identify a, b, and c, where c is the largest side. Here, c = 12 mi, and a and b can be 6 mi and 7 mi respectively.

Apply the Pythagorean Theorem

A triangle is a right triangle if it satisfies the Pythagorean theorem: \(a^2 + b^2 = c^2\). Let's compute \(a^2 + b^2\) and \(c^2\): \(a^2 + b^2 = 6^2 + 7^2 = 36 + 49 = 85\). Now calculate \(c^2 = 12^2 = 144\).

Compare the sums

Compare the sum of the squares of the smaller sides with the square of the largest side. Since \(85 eq 144\), the sides do not satisfy the Pythagorean theorem.

Conclude the Triangle Type

Since \(a^2 + b^2 eq c^2\), the triangle with sides 6 mi, 7 mi, and 12 mi is not a right triangle.

Key concepts

Pythagorean theorem

The Pythagorean Theorem is a fundamental principle in geometry used to determine if a triangle is right-angled. A right triangle features a 90-degree angle. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as:\[ a^2 + b^2 = c^2 \]Where:

• \(a\) and \(b\) are the lengths of the two shorter sides.
• \(c\) is the length of the hypotenuse.

To apply the Pythagorean Theorem to determine if a triangle with sides of 6 mi, 7 mi, and 12 mi is a right triangle, calculate \(a^2 + b^2\) and compare it to \(c^2\). If they match, the triangle is a right triangle; if not, it isn't.

Triangle classification

Triangles can be sorted into distinct categories based on their side lengths and angles. The primary classifications are:

• Equilateral Triangle - All sides and angles are equal.
• Isosceles Triangle - It has at least two equal sides.
• Scalene Triangle - All sides and angles are different.
• Right Triangle - Contains one 90-degree angle.

For a triangle to be classified as a right triangle, it must satisfy the Pythagorean Theorem. The longest side is typically the hypotenuse. In the example of the 6 mi, 7 mi, and 12 mi triangle, because \(a^2 + b^2 eq c^2\), it remains scalene and isn't right-angled.

Geometry problem-solving

In solving geometry problems, especially those involving triangles, it’s essential to follow a clear methodical approach. To determine if a triangle is a right triangle:

1. Identify the longest side as it will be the hypotenuse if the triangle is right-angled.
2. Use the Pythagorean Theorem to check if the relationship \( a^2 + b^2 = c^2 \) holds for the sides given.
3. Compare the side calculations to determine the triangle type.

This step-by-step approach ensures clarity and accuracy. Consider every possible scenario, such as different side combinations, to verify findings. In the referenced problem, comparing \(85\) and \(144\) shows the triangle isn't right-angled. Understanding these steps reinforces core geometry problem-solving skills and aids in solving similar problems efficiently.