Final answer:
Triangle ABC is isosceles because it has two equal sides (AB and BC). It is not a right triangle, equilateral, or equiangular. The correct option for ∆ABC's features is that it is isosceles and scalene.
Step-by-step explanation:
Identifying the Features of ∆ABC
To determine the features of triangle ABC (or ∆ABC), we must examine the given coordinates of its vertices: A(3, 3), B(4, 7), and C(8, 6). By calculating the distances between these points, we can identify the type of triangle and its characteristics.
Step 1: Calculate the distance between each pair of points to determine the lengths of the sides of ∆ABC.
• AB can be found using the distance formula, resulting in AB = √[(4-3)^2 + (7-3)^2] = √[1^2 + 4^2] = √17
• BC can be found similarly, giving us BC = √[(8-4)^2 + (6-7)^2] = √[4^2 + (-1)^2] = √17
• Finally, AC = √[(8-3)^2 + (6-3)^2] = √[5^2 + 3^2] = √34
Since AB = BC, we have two equal sides, which makes ∆ABC isosceles. Also, since AC is not equal to AB or BC, ∆ABC cannot be equilateral.
Step 2: Check for right angles. Using the Pythagorean theorem, we analyze the lengths to see if ∆ABC is a right triangle.
• If ∆ABC were right, then AB^2 + BC^2 should equal AC^2.
• However, (√17)^2 + (√17)^2 does not equal (√34)^2.
Thus, ∆ABC is not a right triangle, and it is also not equiangular because for a triangle to be equiangular, it must be equilateral, which ∆ABC is not.
Since ∆ABC has no equal angles and only two equal sides, the correct option is:
• B. It is isosceles
• D. It is scalene (since it does not have all sides equal)