Question

In the given diagram, ABCDEF is a regular hexagon inscribed in a circle with centre O. PQ is a tangent to the circle at D. Find the value of:

1. ∠FAG 
2. ∠BСD 
3. ∠PDE

Answer 1

To find the values for these angles, we can use the properties of a regular hexagon and basic circle geometry theorems.

In a regular hexagon:

Each interior angle is 120°.

Each exterior angle is 60°.

Each side subtends an angle of 60 at the centre `O (360^circ/6 = 60^circ)`.

a. Find ∠FAG

Assuming G is a point on the line BA extended beyond A, ∠FAG is an exterior angle of the regular hexagon.

The interior angle ∠FAG = 120°.

Since GAB is a straight line,

∠FAG = 180° – 120°

= 60°

b. Find ∠BCD

∠BCD is simply one of the interior angles of the regular hexagon ABCDEF.

Using the formula for interior angles:

`((n - 2) xx 180^circ)/n`

= `((6 - 2) xx 180^circ)/6`

= 120°

c. Find ∠PDE

To solve this, we use the Alternate Segment Theorem, which states that the angle between a tangent (PQ) and a chord (DE) is equal to the angle subtended by that chord in the alternate segment which is ∠DFE.

1. Each side of a regular hexagon subtends 60° at the center.

So, ∠EOD = 60°.

2. The angle subtended by an arc at the circumference is half the angle subtended at the center.

Therefore, `∠DFE = 1/2 xx 60^circ = 30^circ`.

3. By the Alternate Segment Theorem, ∠PDE = ∠DFE.