Name: Sum of Exterior Angles of a Polygon
Uploaded: 2019-07-19
Duration: 5 min 7 s
Description: Everyone talks about the interior angles of a polygon, but what about the external ones? In this video, you will learn about the sum of exterior angles of a...

Question

A B C

is a right-angled triangle and

O

is the mid-point of the side opposite to the right angle. Explain why

O

is equidistant from

A, B

and

C

. (The dotted lines are drawn additionally to help you.)

Accepted Answer

Given diagram is:

Given a right-angled triangle $A B C$ and $O$ is the mid-point of $A C .$

Now, draw a line from $A$ parallel to $B C$ and from $C$ parallel to $B A$ .

Let, the point of intersection of these lines be $D$ . Now Join $O D$ .

Now, in quadrilateral $A B C D$

$A B ∥ D C$ and $B C ∥ A D$

$\Rightarrow$ opposite sides are parallel

$∴ A B C D$ is a parallelogram

We know that,

Adjacent angles of a parallelogram are supplementary

$∠ B + ∠ C = 180^{o}$

$\Rightarrow 90 ° + ∠ C = 180 °$

$\Rightarrow ∠ C = 180 ° - 90 °$

$\Rightarrow ∠ C = 90 °$

Also, opposite angles of a parallelogram are equal.

$∠ A = ∠ C$

$\Rightarrow ∠ A = 90 °$

And $∠ D = ∠ B$

$\Rightarrow ∠ D = 90 °$

Therefore, $∠ A = ∠ B = ∠ C = ∠ D = 90 °$

$\Rightarrow$ Each angle of $A B C D$ is a right angle.

So, $A B C D$ is a parallelogram with all angles $90 °$

$∴ A B C D$ is a rectangle

We know that

The diagonals of a rectangle bisect each other

$O A = O C = \frac{1}{2} A C$ … (1)

$O B = O D = \frac{1}{2} B D$ … (2)

Also, The diagonals of a rectangle are equal in length.

$B D = A C$

Dividing both sides by $2$

$\Rightarrow \frac{1}{2} B D = \frac{1}{2} A C$

$\Rightarrow O B = O A$ (from (1) and (2))

$∴ O A = O B = O C$

Hence, $O$ is equidistant from $A, B$ and $C .$

Jagat Jitendra Matto Solutions for Chapter: Understanding Quadrilaterals, Exercise 4: EXERCISE 12.4

Jagat Jitendra Matto Mathematics Solutions for Exercise - Jagat Jitendra Matto Solutions for Chapter: Understanding Quadrilaterals, Exercise 4: EXERCISE 12.4

Questions from Jagat Jitendra Matto Solutions for Chapter: Understanding Quadrilaterals, Exercise 4: EXERCISE 12.4 with Hints & Solutions

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