Final answer:
The triangle formed by the points A(3,0), B(2,7), and C(6,4) is an isosceles triangle with side lengths of √50, 5, and 5.
Step-by-step explanation:
The given question asks for the name of the triangle formed by the points A(3,0), B(2,7), and C(6,4) and asks for the proof of the name and side lengths. To identify the name of the triangle, we can calculate the lengths of the three sides using the distance formula. Once the side lengths are known, we can classify the triangle based on those lengths.
The distance between A and B is √[(2-3)^2 + (7-0)^2] = √[1^2 + 7^2] = √50. The distance between B and C is √[(6-2)^2 + (4-7)^2] = √[4^2 + (-3)^2] = √25 = 5. The distance between C and A is √[(3-6)^2 + (0-4)^2] = √[(-3)^2 + (-4)^2] = √25 = 5.
Based on these side lengths, we can conclude that the triangle ABC is an isosceles triangle, since two of its sides have the same length.