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In a right triangle ABC, right-angled at C, M is the midpoint of hypotenuse AB, C is joined to M and produced to a point D such that DM = CM. If point D is joined to point B (see figure), then which of the following is true?

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An Intiative by Sri Chaitanya

CM = 12 AB

CM = AB

CM = 2 AB

CM = 14 AB

answer is A.

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Detailed Solution

Given that,
In a right triangle ABC, right angled at C, M is the midpoint of hypotenuse AB, C is joined to M and produced to a point D such that DM = CM.
Question Image

In ΔAMC and ΔBMD,

A M = B M ∵ M is the midpoint of AB

∠ A M C = ∠ B M D Vertically opposite angles

C M = D M Given

Thus, using the SAS congruence rule,

We know that corresponding parts of congruent triangles are equal.

….(i)

Since alternate angles are equal, therefore using the property, we have,

A C | | B D

It is given that ABC is a right triangle, right angled at C.

⇒ ∠ A C B = 90 ° (2)

Using the property of co-interior angles,
Question Image

In ΔACB and ΔCBD,
BD=AC (from (i))
BC=BC (common)

∠ D B C = ∠ A C B = 90 ∘

Thus, using the SAS congruence rule,

We know that corresponding parts of congruent triangles are equal.

Multiply both sides with

12 .

It is given that M is the midpoint of AB and CD.

Thus, by using the equation (1) and (2), we have,

Therefore, we have

Hence, option (1) is correct.

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