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  • In a triangle ABC, D is the mid-point of side AC such that BD equal to AC. Show that angle ABC is a right angle.

In a triangle ABC, D is the mid-point of side AC such that BD is equal to AC. Show that angle ABC is a right angle.

Answers (1)

ABC is a triangle and D is the mid-point at AC

BD = AC

To prove : \angleABC = 90°

Proof : AD = CD = ½ AC       (\because D is mid-point)

BD = AC                                 (given)

So, AD = BD = CD

Let AD = BD

\angleBAD = \angleABD                     (angles opposite to equal sides are equal)

Now, CD = BD

\angleBCD = \angleCBD                      (angles opposite to equal sides are equal)

In \triangleABC,

\angleABC + \angleBAC + \angleBCA = 180° (angle sum property)

\angleABC + \angleBAD + \angleBCD = 180°

Now, \angleABC + \angleABD + \angleCBD = 180°         (\because\angleBAD = \angleABD, \angleBCD = \angleCBD)

Then  \angleABC + \angleABC = 180°                         (\because \angleABD + \angleCBD = \angleABC)

\Rightarrow \angleABC = 90°

Hence, proved.

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