Question

connecting concepts in the diagram, $overline{ab}congoverline{bc}$, $overline{ac}congoverline{cd}$, and $ad = 12$. find the lengths of all segments in the diagram.

Explanation:

Step1: Analyze equal - length segments

Since $\overline{AB}\cong\overline{BC}$, let $AB = BC=x$. Then $AC=AB + BC = 2x$. Also, since $\overline{AC}\cong\overline{CD}$, $CD = AC=2x$.

Step2: Use the length of $AD$

We know that $AD=AC + CD$. Given $AD = 12$, and $AC = CD = 2x$, then $AD=2x+2x=4x$.

Step3: Solve for $x$

Set up the equation $4x = 12$. Divide both sides by 4: $x=\frac{12}{4}=3$.

Step4: Find lengths of all segments

$AB=x = 3$, $BC=x = 3$, $AC=2x = 6$, $CD=2x = 6$.

Answer:

$AB = 3$, $BC = 3$, $AC = 6$, $CD = 6$