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State true or false.

In the given figure, $O$ is the midpoint of each of the line segments $A B$ and $C D$ . Then $A C = B D$ and $A C | | B D$ .

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True

False

answer is A.

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Detailed Solution

It is given that

O

is the midpoint of each of the line segments

A B

and

C D

.
We have to show that,

A C = B D

and

A C | | B D

.
Consider the given figure.
Question Image

From the figure,

A O = O B

and

O D = O C

.
Also, the line segments

AB

and

CD

are concurrent.

A O = O B, O D = O C

(since O is the midpoint)
Consider

Δ A O C

and

Δ B O D

A O = O B

∠ A O C = ∠ B O D

(Vertically opposite angles are equal)

O C = O D

Therefore, by SAS property of congruence,

Δ A O C ≅ Δ B O D

...... (1)
By corresponding part of congruent triangle (C.P.C.T.) rule,

A C = B D

∠ A C O = ∠ B D O

Since,

∠ A C O

and

∠ B D O

are alternate angles.
Therefore,

A C ∥ B D

.
If

O

is the midpoint of each of the line segments

A B

and

C D

.
Then

A C = B D

and

A C | | B D

.
Hence the statement is true.
Hence option 1 is correct.

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