Question

?
\\( \frac { a c } { x z } = \frac { \sqrt { 5 2 } } { \sqrt { 1 3 } } \\)
\\( \delta a b c \sim \delta x y z \\) by the sss similarity
theorem.
which mathematical statement is missing?
\\( \frac { y z } { b c } = \frac { 6 } { 3 } \\)
\\( \angle b \cong \angle y \\)
\\( \frac { b c } { y z } = \frac { 6 } { 3 } \\)
\\( \angle b \cong \angle z \\)

Explanation:

Step1: Recall SSS similarity theorem

SSS similarity theorem states that if the ratios of the corresponding sides of two triangles are equal, then the triangles are similar.

Step2: Analyze side - side ratios

For \(\triangle ABC\) and \(\triangle XYZ\) to be similar by SSS similarity theorem, all three ratios of corresponding sides must be equal. We already have \(\frac{AC}{XZ}=\frac{\sqrt{52}}{\sqrt{13}}\). Now, if we calculate the lengths of \(BC\) and \(YZ\) (using distance formula for coordinates or counting units in the grid - assume \(BC = 6\) units and \(YZ=3\) units), then the ratio \(\frac{BC}{YZ}=\frac{6}{3}\)

Answer:

\(\frac{BC}{YZ}=\frac{6}{3}\)