Question

A manufacturer cuts a piece of metal for a microscope. The resulting piece of metal can be represented in a coordinate plane by a triangle with vertices A(0,0), B(4,10) , and C(8,0). One unit in the coordinate plane represents one millimeter. Is △ ABC is isosceles? Explain your reasoning.

Answer 1

By calculating the side lengths of triangle ABC using the distance formula, it was determined that sides AB and BC are equal in length, while side CA is a different length, confirming that △ABC is an isosceles triangle.

Step-by-step explanation:

To determine whether △ABC is isosceles, we need to compare the lengths of its sides. If two sides are equal in length, the triangle is isosceles. We can use the distance formula, which is derived from the Pythagorean theorem, to calculate the lengths of the sides AB, BC, and CA. The distance formula is √((x2-x1)² + (y2-y1)²).

For segment AB, we have A(0,0) and B(4,10), which gives us the distance AB = √((4-0)² + (10-0)²) = √(16+100) = √116.

For segment BC, we have B(4,10) and C(8,0), which gives us the distance BC = √((8-4)² + (0-10)²) = √(16+100) = √116.

For segment CA, we have C(8,0) and A(0,0), which gives us the distance CA = √((0-8)² + (0-0)²) = √64 = 8 mm.

Since AB = BC, both measuring √116 mm, and CA is different at 8 mm, we can conclude that △ABC is indeed isosceles as it has two sides of equal length.