Question

drag the tiles to the boxes to form correct pairs. match each set of vertices to the triangle they form. right scalene acute equilateral acute isosceles right isosceles obtuse scalene a(2, 4), b(4, 5), c(3, 6) a(3, 5), b(3, 4), c(5, 4) a(3, 5), b(5, 6), c(3, 0) a(2, 4), b(3, 5), c(2, 6)

Explanation:

To solve this, we use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the side lengths of each triangle, then classify them by side lengths (isosceles, scalene, equilateral) and angles (right, acute, obtuse) using the Pythagorean theorem (for right angles) and angle sum/difference.

Step 1: Analyze \(A(2,4), B(4,5), C(3,6)\)

• \(AB: \sqrt{(4 - 2)^2 + (5 - 4)^2} = \sqrt{4 + 1} = \sqrt{5}\)
• \(BC: \sqrt{(3 - 4)^2 + (6 - 5)^2} = \sqrt{1 + 1} = \sqrt{2}\)
• \(AC: \sqrt{(3 - 2)^2 + (6 - 4)^2} = \sqrt{1 + 4} = \sqrt{5}\)

Two sides equal (\(AB = AC\)), check angles. Using dot product or Pythagorean: \((\sqrt{2})^2 + (\sqrt{5})^2 = 2 + 5 = 7\), \((\sqrt{5})^2 = 5\). Since \(7>5\), acute. So acute isosceles.

Step 2: Analyze \(A(3,5), B(3,4), C(5,4)\)

• \(AB: \sqrt{(3 - 3)^2 + (4 - 5)^2} = 1\)
• \(BC: \sqrt{(5 - 3)^2 + (4 - 4)^2} = 2\)
• \(AC: \sqrt{(5 - 3)^2 + (4 - 5)^2} = \sqrt{4 + 1} = \sqrt{5}\)

Check Pythagorean: \(1^2 + 2^2 = 1 + 4 = 5 = (\sqrt{5})^2\). Right angle, all sides different. So right scalene.

Step 3: Analyze \(A(3,5), B(5,6), C(3,0)\)

• \(AB: \sqrt{(5 - 3)^2 + (6 - 5)^2} = \sqrt{4 + 1} = \sqrt{5}\)
• \(BC: \sqrt{(3 - 5)^2 + (0 - 6)^2} = \sqrt{4 + 36} = \sqrt{40}\)
• \(AC: \sqrt{(3 - 3)^2 + (0 - 5)^2} = 5\)

Check Pythagorean: \(5^2 + (\sqrt{5})^2 = 25 + 5 = 30\), \((\sqrt{40})^2 = 40\). Since \(30 < 40\), obtuse. All sides different. So obtuse scalene.

Step 4: Analyze \(A(2,4), B(3,5), C(2,6)\)

• \(AB: \sqrt{(3 - 2)^2 + (5 - 4)^2} = \sqrt{1 + 1} = \sqrt{2}\)
• \(BC: \sqrt{(2 - 3)^2 + (6 - 5)^2} = \sqrt{1 + 1} = \sqrt{2}\)
• \(AC: \sqrt{(2 - 2)^2 + (6 - 4)^2} = 2\)

Two sides equal (\(AB = BC\)), check Pythagorean: \((\sqrt{2})^2 + (\sqrt{2})^2 = 2 + 2 = 4 = 2^2\). Right angle. So right isosceles.

(Note: The "acute equilateral" is not matched here as no triangle has all sides equal with these coordinates.)

Answer:

• \(A(2,4), B(4,5), C(3,6)\) → acute isosceles
• \(A(3,5), B(3,4), C(5,4)\) → right scalene
• \(A(3,5), B(5,6), C(3,0)\) → obtuse scalene
• \(A(2,4), B(3,5), C(2,6)\) → right isosceles