In A B C, If A D E= B, then prove that A D E A B C. Also if A D=7… - Vidyadip

Question

In $\triangle A B C$, If $\angle A D E=\angle B$, then prove that $\triangle A D E \sim \triangle A B C$.
Also if $A D=7.6 cm, A E=7.2 cm, B E=4.2 cm$ and $B C=5.4 cm$, then find $D E$.

✓

Answer

In $\triangle A D E$ and $\triangle A B C$
Given,
$
\begin{array}{ll}
\angle A D E=\angle B & (\text { Given }) \\
\angle A=\angle A & \text { (Common) } \\
\angle C=\angle E & \left(\text { each } 90^{\circ}\right)
\end{array}
$
Hence, $\triangle A D E \sim \triangle A B C$ by R.H.S criterion.
Now $\triangle A D E \sim \triangle A B C$
So,
$
\begin{array}{l}
\frac{A C}{A E}=\frac{A B}{A D}=\frac{B C}{D E} \\
A B=B E+A E \\
\Rightarrow A B=4.2+7.2=11.4 cm
\end{array}
$
Now, $\frac{A C}{A E}=\frac{A B}{A D}=\frac{B C}{D E}$
$
\Rightarrow \frac{A C}{7.2}=\frac{11.4}{7.6}=\frac{5.4}{D E}
$
Now, $\Rightarrow \frac{11.4}{7.6}=\frac{5.4}{D E}$
$
\begin{array}{l}
\Rightarrow 11.4 \times D E=5.4 \times 7.6 \\
\Rightarrow D E=\frac{7.6 \times 5.4}{11.4}
\end{array}
$
Hence $D E=3.6 cm$.

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