Question

The lengths of three sides of a triangle are 4 feet, 6 feet, and 9 feet. Is the triangle a right triangle? (Lesson 6-6)

Short answer

No, the triangle is not a right triangle.

Step-by-step solution

Identify Triangle Sides

Identify the sides of the triangle: let's denote the longest side as the hypothesized hypotenuse, which is 9 feet, and the other two sides as 4 feet and 6 feet.

Apply the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (\[c\]) is equal to the sum of the squares of the lengths of the other two sides (\[a\] and \[b\]): \[a^2 + b^2 = c^2\]. Substitute the known values into the equation: \[4^2 + 6^2 = 9^2\].

Calculate the Squares

Calculate the squares of the lengths:\[4^2 = 16\]\[6^2 = 36\]\[9^2 = 81\].

Validate Using the Pythagorean Theorem

Add the squares of the two shorter sides and compare with the square of the hypothesized hypotenuse:\[16 + 36 = 52\]. Since \[52 eq 81\], the given sides do not satisfy the Pythagorean Theorem.

Conclude Type of Triangle

Since the condition for a right triangle, \[a^2 + b^2 = c^2\], is not satisfied, the triangle with sides 4 feet, 6 feet, and 9 feet is not a right triangle.

Key concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees. The defining feature of a right triangle is that it follows the Pythagorean Theorem. This theorem provides a relationship between the three sides of the triangle: the two legs and the hypotenuse. The hypotenuse is always the longest side, opposite the right angle. In the context of determining whether a triangle is a right triangle, it's important to remember that not all sets of side lengths will satisfy this condition. For example, when given triangle sides, you can apply the Pythagorean Theorem to check if they indeed form a right triangle. This requires understanding how to manipulate and solve basic algebraic equations.

Triangle Sides

Triangles are basic geometrical shapes, and each triangle has three sides. In a right triangle, these sides are commonly referred to as:

• Legs: These are the two sides that form the right angle.
• Hypotenuse: This is the side opposite the right angle and is the longest side of the triangle.

When analyzing triangle sides, especially in relation to finding if it's a right triangle, ensure you correctly identify these parts. By taking the largest numerical value as the potential hypotenuse, you can test the provided side lengths with the Pythagorean Theorem. This allows you to see if there's a right angle present.

Triangle Inequalities

Triangle inequalities are foundational rules in geometry that every triangle must satisfy. A key concept here is that the sum of any two sides of a triangle must always be greater than the third side. For our task, it's crucial to ensure the side lengths given could even form a triangle, irrespective of the right angle determination. Specifically:

• The side lengths should be checked such that 4 + 6 > 9, 4 + 9 > 6, and 6 + 9 > 4 hold true.
• If any combination of side lengths fails this test, then a triangle cannot be formed. However, in our problem, the inequality is satisfied, meaning a triangle can exist, but not necessarily a right one.

Understanding these inequalities helps avoid misinterpretations when checking for potential right triangles.