Question

Theorem: In a right angled triangle, if the altitude is drawn to the hypotenuse, then
the two triangles formed are similar to the original triangle and to each
other.
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Answer 1

Step-by-step explanation:

you have to congruency between all the triangles formed then ull get the answer

Answer 2

We need to recall the following properties for similar triangles.

Two triangles are said to be similar if,

• Their corresponding angles are equal.
• Their corresponding sides are proportional.

AAA or AA Similarity: If two triangles are equiangular to one another, then the triangles are similar.

This problem is about similar triangles.

Given:

The altitude is drawn to the hypotenuse of a right-angled triangle.

To Prove: The two triangles formed are similar to the original triangle and to each other.

In $\triangle PQR$  and  $\triangle PSQ$,

$\angle PQR=\angle PSQ=90\textdegree$

$\angle QPR=\angle QPS$                 .......(Common angle)

So, by AA similarity, we get

$\triangle PQR$ ~ $\triangle PSQ$                ........$(1)$

In $\triangle PQR$  and  $\triangle QSR$,

$\angle PQR=\angle QSR=90\textdegree$

$\angle PRQ=\angle QRS$                 .......(Common angle)

So, by AA similarity, we get

$\triangle PQR$ ~ $\triangle QSR$                ........$(2)$

From the equations $(1)$ and $(2)$, we get

$\triangle PQR$ ~ $\triangle PSQ$ ~ $\triangle QSR$  

Hence, proved.