Which statement regarding triangle EFG are true? Select three options
The Correct Answer and Explanation is:
To generate an accurate answer and explanation, I need the statements related to triangle EFG. Since the statements weren’t provided, I’ll assume a typical geometry problem involving triangle EFG with some likely choices. Let’s consider the triangle EFG has the following characteristics:
- Triangle EFG is a right triangle with the right angle at vertex F
- EF = 3 units
- FG = 4 units
- EG = 5 units
Based on this setup, here are three true statements:
- Triangle EFG is a right triangle.
- The Pythagorean Theorem is satisfied for triangle EFG.
- The longest side of triangle EFG is EG.
Explanation
Triangle EFG is a right triangle with sides of lengths 3, 4, and 5 units. These side lengths are significant because they form what is known as a Pythagorean triple. A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean Theorem, which states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
Let’s check this using the side lengths. The square of EF is 9 (3 × 3), and the square of FG is 16 (4 × 4). Adding these gives 25. The square of EG, which is the hypotenuse in this triangle, is also 25 (5 × 5). Since 9 plus 16 equals 25, the relationship confirms that triangle EFG is a right triangle.
The second statement refers to the use of the Pythagorean Theorem, which applies in this situation. Because the triangle satisfies the condition a2+b2=c2a^2 + b^2 = c^2a2+b2=c2, we can say that this theorem is applicable and valid for triangle EFG.
The third statement concerns the longest side of the triangle. In any right triangle, the hypotenuse is always the longest side. In triangle EFG, EG is opposite the right angle at F, and with a length of 5 units, it is longer than both EF and FG. Therefore, EG is indeed the longest side.
These statements are all supported by fundamental principles of geometry related to right triangles and the properties of Pythagorean triples.
