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: Recall the distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate the length of AB

For points $A(3,5)$ and $B(5,7)$, $x_1 = 3,y_1 = 5,x_2 = 5,y_2 = 7$. Then $AB=\sqrt{(5 - 3)^2+(7 - 5)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$ units.

Step3: Calculate the length of BC

For points $B(5,7)$ and $C(7,5)$, $x_1 = 5,y_1 = 7,x_2 = 7,y_2 = 5$. Then $BC=\sqrt{(7 - 5)^2+(5 - 7)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$ units.

Step4: Calculate the length of AC

For points $A(3,5)$ and $C(7,5)$, $x_1 = 3,y_1 = 5,x_2 = 7,y_2 = 5$. Then $AC=\sqrt{(7 - 3)^2+(5 - 5)^2}=\sqrt{16+0}=4$ units.

Step5: Calculate the perimeter of $\triangle ABC$

The perimeter $P=AB + BC+AC=2\sqrt{2}+2\sqrt{2}+4 = 4\sqrt{2}+4
eq12$ units. So 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.