Question

The diameter of hemisphere makes the following angles. Explain?

(A) 45°

(B) 180°

(C) 90°

(D) 60°
​

Answer 1

Step-by-step explanation:

This is a known and a very useful property of inscribed angles that they measure half the central angle subtended by the same arc, or, which is the same, by the same chord. When a chord is a diameter, the central angles measures _________

right angles and the corresponding inscribed angles are all _________

. The applet above specifically demonstrates this fact, which is sufficiently important to warrant an independent proof. (The statement is often referred to as Thales' theorem.)

Let P be a point on a circle with diameter AB and center O. So that OA = OB = OP, as three radii of the same circle. This makes triangles AOP and BOP _________

. In each, the base angles are equal and their sum equals the opposite exterior angle:

∠OAP + ∠APO = ∠ _________

and also

∠OBP + ∠BPO = ∠ _________

.

But since ∠OAP = ∠ _________

and ∠OBP = ∠BPO, we further have

2∠APO = ∠ _________

and also

2∠BPO = ∠ _________

.

Adding the two up

2∠ _________

+ 2∠BPO = ∠AOP + ∠ _________

= 180°.

In other words,

∠APB = ∠ _________

+ ∠BPO = 90°.

Q.E.D.