Question

1. triangle abc has vertices at a(3,5), b(5,7) and c(7,5). lisa believes that the perimeter of △abc is 12 units. why is lisa incorrect? a. because ab = 2√2 units, bc = 2√2 units and ac = 4 units. b. because ab = 8 units, bc = 8 units and ac = 16 units. c. because ab = 2 units, bc = 2 units and ac = 2 units. d. because ab = 4 units, bc = 4 units and ac = 8 units.

Explanation:

Step1: Use distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$

For $AB$ with $A(3,5)$ and $B(5,7)$: $AB=\sqrt{(5 - 3)^2+(7 - 5)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$

Step2: For $BC$ with $B(5,7)$ and $C(7,5)$: $BC=\sqrt{(7 - 5)^2+(5 - 7)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$

Step3: For $AC$ with $A(3,5)$ and $C(7,5)$: $AC=\sqrt{(7 - 3)^2+(5 - 5)^2}=\sqrt{16}=4$

Step4: Calculate perimeter

Perimeter of $\triangle ABC=AB + BC+AC=2\sqrt{2}+2\sqrt{2}+4 = 4 + 4\sqrt{2}
eq12$. Lisa is incorrect because $AB = 2\sqrt{2}$ units, $BC = 2\sqrt{2}$ units and $AC = 4$ units.

Answer:

A. Because $AB = 2\sqrt{2}$ units, $BC = 2\sqrt{2}$ units and $AC = 4$ units.