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State true or false.

In the given figure, $O$ is the center of the circle and $X O Y$ is a diameter. If $X Z$ is any other chord of the circle, then $X Y > X Z$ .

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True

False

answer is A.

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Detailed Solution

It is given that

O

is the center of the circle and

X O Y

is a diameter.

X Z

is any other chord of the circle.
We have to show that,

X Y > X Z

.
Consider the figure given below-
Question Image

Consider

Δ X O Z

.
Since, the sum of any

2

sides in a triangle is greater than its third side.

O X + O Z > X Z

……(1)
The radius of the circle,

O Z = O Y

......(2)
Diameter of the circle,

O X + O Y = X Y

......(3)
Consider

Δ X O Z

and equation (1).

O X + O Z > X Z

O X + O Y > X Z

from equation (2),
Therefore,

X Y > X Z

from equation (3).
If

O

is the center of the circle and

X O Y

is a diameter. If

X Z

is any other chord of the circle, then

X Y > X Z

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

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