Question

In the given figure, STI|PQ, U and T are respectively the mid-points of the sides RS and RQ. Prove that RS^2 = PR× RU.

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Answer 1

Answer:

PR*RU=PR²U

Step-by-step explanation:

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Answer 2

Given : $\[ST||PQ\]$, U and T are respectively the mid-points of the sides $RS$and $PQ$.

To Find : Prove that $\[R{S^2} = PR \times RU\]$

Solution:

Understand that from the question,

$\[ST||PQ\]$

$\[ \Rightarrow \Delta RST\~\Delta RPQ\]$ ~$\[\Delta RPQ\]$  by AA  similarity  as $\[\angle S = \angle P\]$ and $\[\angle T = \angle Q\]$ ( corresponding angles)

$\[ \Rightarrow \frac{{RS}}{{RP}} = \frac{{RT}}{{TQ}}\]$------(1)

$\[ \Rightarrow \frac{{RS}}{{RP}} = \frac{{RU}}{{RS}}\]$

$\[\begin{array}{l} \Rightarrow R{S^2} = RP \times RU\\ \Rightarrow R{S^2} = PR \times RU\end{array}\]$

Observe that,  U and T are respectively the mid-points of the sides $RS$ and $PQ.$

$\[ \Rightarrow \Delta RUT\~\Delta RSQ\]$~$\[\Delta RSQ\]$

$\[ \Rightarrow \frac{{RU}}{{RS}} = \frac{{RT}}{{TQ}}\]$------(2)

Understand that from equation (1) and (2).

$\[\begin{array}{l} \Rightarrow \frac{{RS}}{{RP}} = \frac{{RU}}{{RS}}\\ \Rightarrow R{S^2} = RP \times RU\end{array}\]$

 Hence proved