Question

In the figure below ad and be are diameters of circle p. what is the arc measure of cd in degrees

Answer 1

Given:

In the figure below ad and be are diameters of circle p.

To find:

What is the arc measure of cd in degrees ?

Solution:

Consider the attached required figure, while going through the following steps.

From given figure, we have,

Given is a circle, as we know that sum of interior angles of a circle equals 360°, therefore, we have,

∠ APB + ∠ BPC + ∠ CPD + ∠ DPE + ∠ EPA = 360°

but, ∠ EPA  + ∠ APB = ∠ BPD  + ∠ DPE = 180° (∵ they form straight lines)

As, we need to find arc CD, consider ∠ BPD  + ∠ DPE = 180°

again, ∠ BPD  = ∠ DPE = 180°/2 = 90°       (from figure)

∠ DPE = 90° = (33k - 9)°

⇒ 90° = (33k - 9)°

⇒ 90° + 9° = 33k

⇒ 99° = 33k

∴ k = 3°

Now consider, ∠ CPD = (20k + 4)°

⇒ ∠ CPD = (20 × 3 + 4)°

⇒ ∠ CPD = 64°

w.k.t the measure of the  intercepted arc is twice that of the inscribed angle.

∠ = 1/2 (m arc)

∠ CPD = 1/2 (m arc CD)

64° = 1/2 (m arc CD)

64° × 2 = m arc CD

∴ m arc CD = 128°