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What does the angle in a semicircle measure?

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45 ∘

60 ∘

90 ∘

36 ∘

answer is C.

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Detailed Solution

We have to find the measure of the angle in a semi-circle.
Question Image

The angle in a semicircle is

∠ B A C

.
From the figure,
OA=OB=OC (Radius of the same circle)
So, OA=OB
Since ∆AOB is an isosceles triangle,

∴ ∠ A B O = ∠ O A B

Again, OA=OC
Since ∆AOC is an isosceles triangle,

∴ ∠ A C O = ∠ O A C

In ∆ABC,

∠ A B C + ∠ B A C + ∠ A C B = 1800 ⇒ ∠ A B C + ∠ O A B + ∠ O A C + ∠ A C B = 1800 ⇒ ∠ A B C + ∠ A B O + ∠ A C O + ∠ A C B = 1800 ⇒ 2 × (∠ A B O + ∠ A C O) = 1800 [∵ ∠ A B C = ∠ A B O, ∠ A C B = ∠ A C O] ⇒ (∠ A B O + ∠ A C O) = 18002 ⇒ (∠ A B O + ∠ A C O) = 900

Observing the figure,

∠ B A C = ∠ O A B + ∠ O A C = ∠ A B O + ∠ A C O = 900

The value of the angle in a semicircle is

90 ∘

.
Therefore, option 3 is correct.

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