Question

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

Explanation:

Step1: Recall Distance Formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). We'll use this to find the side lengths of each triangle.

Step2: Analyze Triangle \(A(2,4), B(3,5), C(2,6)\)

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

Two sides (\(AB = BC=\sqrt{2}\)) are equal. Check angles: Slope of \(AC\) is undefined (vertical line), slope of \(AB\) is \(\frac{5 - 4}{3 - 2}=1\), slope of \(BC\) is \(\frac{6 - 5}{2 - 3}=-1\). Product of slopes of \(AB\) and \(BC\) is \(-1\), so right angle. Thus, right isosceles.

Step3: Analyze Triangle \(A(3,5), B(5,6), C(3,0)\)

• \(AB\): \(\sqrt{(5 - 3)^2+(6 - 5)^2}=\sqrt{4 + 1}=\sqrt{5}\)
• \(AC\): \(\sqrt{(3 - 3)^2+(0 - 5)^2}=\sqrt{0 + 25}=5\)
• \(BC\): \(\sqrt{(3 - 5)^2+(0 - 6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}\)

All sides different. Check largest side \(BC = 2\sqrt{10}\approx6.32\), \(AB^2+AC^2 = 5 + 25 = 30\), \(BC^2 = 40\). Since \(AB^2+AC^2

Step4: Analyze Triangle \(A(2,4), B(4,5), C(3,6)\)

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

Two sides equal. Check angles: Slope of \(AB\) is \(\frac{5 - 4}{4 - 2}=\frac{1}{2}\), slope of \(AC\) is \(\frac{6 - 4}{3 - 2}=2\). Product of slopes is \(1\), so acute angle. Thus, acute isosceles.

Step5: Analyze Triangle \(A(3,5), B(3,4), C(5,4)\)

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

All sides different. Slope of \(AB\) is undefined (vertical), slope of \(BC\) is \(0\) (horizontal), so right angle. Thus, right scalene.

Answer:

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

(Note: The "acute equilateral" option might be a distractor as no triangle here is equilateral. The fourth triangle's analysis was added for completeness based on typical problem structures, assuming a possible typo or missed vertex set, but following the given vertex sets, the above matches.)