Question

Given: ABC, A(3,6), B(5,0), C(1,0)
Prove: ABC is an isosceles triangle.

Answer 1

To prove triangle ABC is an isosceles triangle, distances AB and AC were calculated and found to be equal using the distance formula derived from the Pythagorean theorem, thus confirming that ABC is an isosceles triangle.

Step-by-step explanation:

To prove that triangle ABC is an isosceles triangle, we need to show that at least two sides of the triangle are of equal length. We calculate the lengths of AB, BC, and AC using the distance formula, which is derived from the Pythagorean theorem:

Distance formula: Given two points P1(x1, y1) and P2(x2, y2), the distance between them is d = \(\sqrt{(x2-x1)^2 + (y2-y1)^2}\). Applying this formula:

1. For AB: \(d = \sqrt{(5-3)^2 + (0-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}\)
2. For BC: \(d = \sqrt{(1-5)^2 + (0-0)^2} = \sqrt{16 + 0} = \sqrt{16} = 4\)
3. For AC: \(d = \sqrt{(1-3)^2 + (0-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}\)

The lengths of AB and AC are both \(2\sqrt{10}\), which means they are equal. Since two sides of the triangle are equal, triangle ABC is indeed an isosceles triangle.