Question

c. $sqrt{(y_2 - y_1)^2 -(x_2 - x_1)^2}$
d. $sqrt{(x_2 + x_1)^2 -(y_2 + y_1)^2}$

question 6
is the following shape a rectangle? how do you know?
a. no, because the figure is not a quadrilateral.
b. there is not enough information to determine.
c. yes, the adjacent sides are perpendicular, and the opposite sides are parallel.
d. no, the adjacent sides are not perpendicular.

question 7
is the following shape a right triangle? how do you know?
a. no, there is no right angle.
b. there is not enough information to determine.
c. no, the side lengths do not fit the pythagorean theorem.
d. yes, two sides are perpendicular, and the side lengths fit the pythagorean theorem.

Explanation:

Step1: Identify Question 6 points

From the graph, assign coordinates: $A(2,2)$, $B(1,6)$, $C(-2,4)$, $D(0,0)$

Step2: Calculate side slopes (Question6)

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
Slope $AB$: $\frac{6-2}{1-2}=\frac{4}{-1}=-4$
Slope $BC$: $\frac{4-6}{-2-1}=\frac{-2}{-3}=\frac{2}{3}$
Slope $CD$: $\frac{0-4}{0-(-2)}=\frac{-4}{2}=-2$
Slope $DA$: $\frac{2-0}{2-0}=1$

Step3: Check perpendicularity (Question6)

Perpendicular slopes multiply to $-1$.
$-4 \times \frac{2}{3}
eq -1$, $\frac{2}{3} \times -2
eq -1$, etc. No adjacent sides are perpendicular.

Step4: Identify Question7 points

From the graph, assign coordinates: $A(0,4)$, $B(2,2)$, $C(-1,0)$

Step5: Calculate side slopes (Question7)

Slope $AB$: $\frac{2-4}{2-0}=\frac{-2}{2}=-1$
Slope $BC$: $\frac{0-2}{-1-2}=\frac{-2}{-3}=\frac{2}{3}$
Slope $CA$: $\frac{4-0}{0-(-1)}=4$

Step6: Check perpendicularity (Question7)

No pair of slopes multiplies to $-1$, so no right angle.

Answer:

Question 6: D. No, the adjacent sides are not perpendicular.
Question 7: A. No, there is no right angle.