Question

the statements below can be used to prove that the triangles are similar.
image of coordinate plane with triangles abc and xyz
$\frac{ab}{xy} = \frac{4}{2}$
?
$\frac{ac}{xz} = \frac{sqrt{52}}{sqrt{13}}$
$\triangle abc sim \triangle xyz$ by the sss similarity theorem.
which mathematical statement is missing?
$\bigcirc \frac{yz}{bc} = \frac{6}{3}$
$\bigcirc angle b cong angle y$
$\bigcirc \frac{bc}{yz} = \frac{6}{3}$
$\bigcirc angle b cong angle z$

Explanation:

Step1: Recall SSS Similarity Theorem

The SSS (Side - Side - Side) similarity theorem states that if the corresponding sides of two triangles are in proportion, then the triangles are similar. So we need to find the ratio of the third pair of corresponding sides.

First, let's find the lengths of the sides of the triangles.

For $\triangle ABC$:

• $AB$: From the graph, $A$ is at $(-4,4)$ and $B$ is at $(-4,0)$, so the length of $AB$ is $|4 - 0|=4$.
• $BC$: $B$ is at $(-4,0)$ and $C$ is at $(2,0)$, so the length of $BC$ is $|2-(-4)| = 6$.
• $AC$: Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, for $A(-4,4)$ and $C(2,0)$, $AC=\sqrt{(2 + 4)^2+(0 - 4)^2}=\sqrt{36 + 16}=\sqrt{52}$.

For $\triangle XYZ$:

• $XY$: $X$ is at $(0,2)$ and $Y$ is at $(0,4)$, so the length of $XY$ is $|4 - 2| = 2$.
• $YZ$: $Y$ is at $(0,4)$ and $Z$ is at $(3,4)$, so the length of $YZ$ is $|3-0|=3$.
• $XZ$: Using the distance formula for $X(0,2)$ and $Z(3,4)$, $XZ=\sqrt{(3 - 0)^2+(4 - 2)^2}=\sqrt{9 + 4}=\sqrt{13}$.

We know that $\frac{AB}{XY}=\frac{4}{2}$ and $\frac{AC}{XZ}=\frac{\sqrt{52}}{\sqrt{13}} = 2$ (since $\sqrt{52}=2\sqrt{13}$, so $\frac{2\sqrt{13}}{\sqrt{13}} = 2$). We need to find the ratio of the third pair of corresponding sides, which should be $\frac{BC}{YZ}$.

Calculate $\frac{BC}{YZ}$: $BC = 6$, $YZ=3$, so $\frac{BC}{YZ}=\frac{6}{3}=2$, which is the same as the other two ratios.

Now let's analyze the options:

• Option 1: $\frac{YZ}{BC}=\frac{6}{3}$ is incorrect because $\frac{YZ}{BC}=\frac{3}{6}=\frac{1}{2}$, not $\frac{6}{3}$.
• Option 2: $\angle B\cong\angle Y$: $\angle B$ and $\angle Y$ are right angles (since $AB\perp BC$ and $XY\perp YZ$), but SSS similarity is about side ratios, not angles (that's SAS or AA). Also, we are using SSS, so this is not the missing statement.
• Option 3: $\frac{BC}{YZ}=\frac{6}{3}$: As we calculated, $BC = 6$, $YZ = 3$, so $\frac{BC}{YZ}=\frac{6}{3}=2$, which is the same ratio as $\frac{AB}{XY}$ and $\frac{AC}{XZ}$, so this satisfies the SSS similarity condition.
• Option 4: $\angle B\cong\angle Z$: $\angle B$ is a right angle, $\angle Z$ is not a right angle (from the triangle $XYZ$, $\angle Y$ is the right angle), so this is incorrect.

Answer:

$\boldsymbol{\frac{BC}{YZ}=\frac{6}{3}}$ (the third option)