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Question

ABC and AMP are two right triangles right angled at B and M respectively. Prove that
i) ABCAMP ii) CAPA=BCMP

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Solution

Given figure : (Image )
i) To prove
ABC is ismilar to triangle AMP
We can prove this using angle-angle similarity postulate
In triangles AMP and ABC
AMP=ABC=90°
MAP=BAC=same angle (common angle)
Hence, two angles of the required triangles are same
By angle-angle (AA) similarity:
We can say that ABCAMP
ii) To prove CAPA=BCMP
In part i) we already proved
that AMP=ABC=90°
and MAP=BAC= common angle
and sum of 3 angles of a triangle=180°
APM=ACB
From side-angle-side similarity theorem,
we can that the sides including the angles which are equal must be proportional to each other
Sides including APM=AP and MP
Sides including ACB=AC & BC
Using S-A-S similarity theorem
APMP=ACBC
BCMP=ACAP
CAPA=BCMP
Hence proved.

1037228_1078608_ans_0659e3960ead4901bc88972aedda8d49.JPG

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