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Find the angle subtended at the origin by the line segment whose end points are A (0,100) and B (10,0).

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90 °

80 °

65 °

70 °

answer is A.

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Detailed Solution

Given points is,

A 0, 100 and B 10, 0

.
Let

O 0, 0

be the origin.
Since,
The distance between any two points in a Cartesian plane

M a, b and N c, d

is given by

M N = (c − a) 2 + (d − b) 2

.
Then,

A B = (10 − 0) 2 + (0 − 100) 2 ⇒ A B = 100 + 10000 ⇒ A B = 10100

A O = (0 − 0) 2 + (0 − 100) 2 ⇒ A O = 1002 ⇒ A O = 100

And

B O = (10 − 0) 2 + (0 − 0) 2 ⇒ B O = 100 ⇒ B O = 10

Now,

A O 2 + B O 2 = 1002 + 102 ⇒ A O 2 + B O 2 = 10000 + 100 ⇒ A O 2 + B O 2 = 10100 and A B 2 = (10100) 2 ⇒ A B 2 = 10100

∴ A O 2 + B O 2 = A B 2

It also satisfied the Pythagorean theorem.
Then,

Δ A B C

is a right-angled triangle.

∴ ∠ O = 90 °

Therefore, the angle subtended at the origin by the line segment is

900

Hence, option (1) is correct.

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