Question

for help with questions 2 and 3, see example 2. 2. a) verify that de and bc are parallel. b) list the other line segments that are parallel. c) verify that de = bf. d) list the other line segments that have equal lengths.

Explanation:

Step1: Recall slope - formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of DE

Let $D(x_1,y_1)$ and $E(x_2,y_2)$. From the graph, assume $D(6,10)$ and $E(10,8)$. Then $m_{DE}=\frac{8 - 10}{10 - 6}=\frac{-2}{4}=-\frac{1}{2}$.

Step3: Calculate slope of BC

For $B(3,5)$ and $C(13,1)$, $m_{BC}=\frac{1 - 5}{13 - 3}=\frac{-4}{10}=-\frac{1}{2}$.
Since $m_{DE}=m_{BC}=-\frac{1}{2}$, DE and BC are parallel.

Step4: Find other parallel line - segments

By observing the equal - spaced markings on the lines, we can see that AD and EF are parallel, and AB and FC are parallel.

Step5: Recall distance formula

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

Step6: Calculate length of DE

For $D(6,10)$ and $E(10,8)$, $DE=\sqrt{(10 - 6)^2+(8 - 10)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$.

Step7: Calculate length of BF

For $B(3,5)$ and $F(7,3)$, $BF=\sqrt{(7 - 3)^2+(3 - 5)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$. So $DE = BF$.

Step8: Find other equal - length line - segments

By using the distance formula: $AD = EF$, $AB=FC$, $BD = AF$, $AE=DC$.

Answer:

a) DE and BC are parallel because their slopes are equal ($m_{DE}=m_{BC}=-\frac{1}{2}$).
b) AD and EF, AB and FC are parallel.
c) DE = BF as both have a length of $\sqrt{20}=2\sqrt{5}$.
d) AD = EF, AB=FC, BD = AF, AE=DC.