Scott said that \( \triangle A B C \) denoted by the coordinates \( A(2,3) B(B, 3) C(5,8) \) is equilateral. is he correc Yes, he is correct, the side lengths are 6 . No, it is an isosceles triangle with legs lengths of 6 No, it is an isosceles triangle with regs lengins of \( \sqrt{34} \). ves, he is correct, the side lengits are \( \sqrt{34} \).

We start with the vertices   A = (2, 3),   B = (B, 3)      (i.e. some point on the horizontal line y = 3),   C = (5, 8). First, notice that A and B lie on the line y = 3, so AB is horizontal and its length is   AB = |B – 2|. Also, compute AC:   AC = √[(5 – 2)² + (8 – 3)²] = √(3² + 5²) = √(9 + 25) = √34. Similarly, the distance from B to C is   BC = √[(5 – B)² + (8 – 3)²] = √[(5 – B)² + 25]. Scott claimed that the triangle is equilateral (all three sides equal). Let’s check whether it is possible. For an equilateral triangle we would need   AB = AC = BC. Since AC = √34, we would need   |B – 2| = √34     (1) and also   √[(5 – B)² + 25] = √34   (2). From (1) we have   B – 2 = ±√34   → B = 2 ± √34. Now, consider equation (2). Square both sides:   (5 – B)² + 25 = 34   → (5 – B)² = 9  → 5 – B = ±3. That gives   B = 5 ∓ 3   so B = 2  or B = 8. Now, compare these two results: • Equation (1) gives B = 2 ± √34. Since √34 ≈ 5.83, the two possibilities are approximately B ≈ 2 + 5.83 = 7.83 or B ≈ 2 – 5.83 = –3.83. • Equation (2) gives B = 2 or B = 8. The only chance for agreement is when B is about 8 (recalling that 7.83 is very close to 8, but not exactly equal to 8 because √34 is irrational). In fact, if we set B = 8 exactly:   AB = |8 – 2| = 6,   AC = √34 (approximately 5.83),   BC = √[(5 – 8)² + 25] = √(9 + 25) = √34. So with B = 8 we get: AB = 6 and AC = BC = √34. Since √34 ≠ 6, we cannot have all three sides equal. That is, the triangle is not equilateral. Instead, with B = 8 the triangle has two equal sides (AC = BC = √34) and a base AB = 6. Hence it is an isosceles triangle with legs of length √34. Looking at the given answer choices, the correct one is:   "No, it is an isosceles triangle with regs lengins of √34." (Here “regs lengins” is read as “legs lengths”.) Therefore, Scott’s claim that the triangle is equilateral is incorrect.