given: right $ \triangle a b d $ and right $ \triangle c b d $
prove: $ \triangle a b d \cong \triangle c b d $
proof:
find the length of $ \overline{a d} $
$ \begin{aligned} a d & = \sqrt{\left(x_{2}-x_{1}\
ight)^{2}+\left(y_{2}-y_{1}\
ight)^{2}} \\ & = \sqrt{(4 - 0)^{2}+(0 - 0)^{2}} \\ & = \sqrt{4^{2}} = \end{aligned} $
find the length of $ \overline{d c} $
$ \begin{aligned} d c & = \sqrt{\left(x_{2}-x_{1}\
ight)^{2}+\left(y_{2}-y_{1}\
ight)^{2}} \\ & = \sqrt{(8 - 4)^{2}+(0 - 0)^{2}} \\ & = \sqrt{4^{2}} = \end{aligned} $
- $ \overline{a d} \cong \overline{d c} $ by the
- $ \overline{b d} \cong \overline{b d} $ by the
- $ \triangle a b d \cong \triangle c b d $ by the f congruence theorem
a. 4
b. reflexive property of $ \cong $
c. segment property
d. 8
e. 16
f. leg leg
g. definition of $ \cong $

Question

Accepted Answer

# Explanation:
## Step1: Calculate length of $ \overline{AD} $
$$
\begin{align*}
AD&=\sqrt{(4 - 0)^2+(0 - 0)^2}\
&=\sqrt{4^2}\
&=4
\end{align*}
$$
## Step2: Calculate length of $ \overline{DC} $
$$
\begin{align*}
DC&=\sqrt{(8 - 4)^2+(0 - 0)^2}\
&=\sqrt{4^2}\
&=4
\end{align*}
$$
Since $ AD = DC=4 $, $ \overline{AD}\cong\overline{DC} $ by the definition of congruent segments ($ \text{Definition of }\cong $).
## Step3: Analyze $ \overline{BD}\cong\overline{BD} $
$ \overline{BD}\cong\overline{BD} $ by the Reflexive Property of $ \cong $ (a segment is congruent to itself).
## Step4: Determine congruence of triangles
For right - triangles $ \triangle ABD $ and $ \triangle CBD $, we have two pairs of congruent legs ($ \overline{AD}\cong\overline{DC} $ and $ \overline{BD}\cong\overline{BD} $). So, $ \triangle ABD\cong\triangle CBD $ by the Leg - Leg ($ LL $) Congruence Theorem for right - triangles.

# Answer:
For the length of $ \overline{AD} $: $ 4 $ (a). For the length of $ \overline{DC} $: $ 4 $ (a). $ \overline{AD}\cong\overline{DC} $ by the $ \text{Definition of }\cong $ (g). $ \overline{BD}\cong\overline{BD} $ by the $ \text{Reflexive Property of }\cong $ (b). $ \triangle ABD\cong\triangle CBD $ by the $ \text{Leg - Leg} $ ($ f $) Congruence Theorem.

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate length of \( \overline{AD} \)

Step2: Calculate length of \( \overline{DC} \)

Step3: Analyze \( \overline{BD}\cong\overline{BD} \)

Step4: Determine congruence of triangles

Answer: