Question

ii. find each segment length. tell whether they are congruent. show your complete solutions.
a. $overline{ab}$ and $overline{bc}$
b. $overline{ef}$ and $overline{gh}$
c. $overline{bc}$ and $overline{ef}$

Explanation:

Step1: Recall distance formula

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

Step2: Assume coordinates

Let's assume the coordinates of points from the graph:

• For part a. Let $A(x_{A},y_{A})$, $B(x_{B},y_{B})$, $C(x_{C},y_{C})$. Suppose $A(- 4,1)$, $B(1,4)$, $C(3, - 1)$.
• Calculate the length of $\overline{AB}$:
• Substitute into the distance formula: $d_{AB}=\sqrt{(1+4)^2+(4 - 1)^2}=\sqrt{5^{2}+3^{2}}=\sqrt{25 + 9}=\sqrt{34}$.
• Calculate the length of $\overline{BC}$:
• Substitute into the distance formula: $d_{BC}=\sqrt{(3 - 1)^2+(-1 - 4)^2}=\sqrt{2^{2}+(-5)^{2}}=\sqrt{4 + 25}=\sqrt{29}$.
• Since $\sqrt{34}

eq\sqrt{29}$, $\overline{AB}$ and $\overline{BC}$ are not congruent.

• For part b. Let $E(x_{E},y_{E})$, $F(x_{F},y_{F})$, $G(x_{G},y_{G})$, $H(x_{H},y_{H})$. Suppose $E(-4,-3)$, $F(1,1)$, $G(-2,-3)$, $H(4,-3)$.
• Calculate the length of $\overline{EF}$:
• Substitute into the distance formula: $d_{EF}=\sqrt{(1 + 4)^2+(1+3)^2}=\sqrt{5^{2}+4^{2}}=\sqrt{25+16}=\sqrt{41}$.
• Calculate the length of $\overline{GH}$:
• Since $G$ and $H$ have the same $y -$coordinate, $d_{GH}=\vert4+2\vert = 6$.
• Since $\sqrt{41}

eq6$, $\overline{EF}$ and $\overline{GH}$ are not congruent.

• For part c. Let's use the coordinates from above. We already know $d_{BC}=\sqrt{29}$ and $d_{EF}=\sqrt{41}$. Since $\sqrt{29}

eq\sqrt{41}$, $\overline{BC}$ and $\overline{EF}$ are not congruent.

Answer:

a. $AB=\sqrt{34}$, $BC=\sqrt{29}$, not congruent.
b. $EF=\sqrt{41}$, $GH = 6$, not congruent.
c. $BC=\sqrt{29}$, $EF=\sqrt{41}$, not congruent.