Question

In the diagram, is the perpendicular bisector of AB and the angle bisector of CPD . 100 POINTS!

Answer 1

Since ∠CPD = x and segment PN is the angle bisector of this angle, therefore segment PN equally divides ∠CPD into two angles. Which means that:

∠CPN = ∠NPD = x / 2

Further, segment PN is also the perpendicular bisector of AB which further means that the intersection formed by PN and AB creates a right angle (90°). Therefore:

∠NPD + ∠DPB = 90°

x/2 + ∠DPB = 90°

∠DPB = 90 – x/2

Therefore:

sin∠DPB = sin(90 – x/2) which is not in the choices

However we know that the relationship of sin and cos is:

sin(π/2 - θ) = cos θ

Where,

π/2 = 90

θ = x/2

Therefore:

sin(90 – x/2) = cos(x/2)

 

Answer:

cos(x/2)

Answer 2

$\huge\mathfrak\red{Answer:-}$

cos (x/2) ✓✓✓✓
$<b>$
Step-by-step explanation:

∠CPD = x and segment PN is the angle bisector of this angle, therefore segment PN equally divides ∠CPD into two angles. Which means that:

∠CPN = ∠NPD = x / 2

Segment PN is also the perpendicular bisector of AB which further means that the intersection formed by PN and AB creates a right angle (90°).
So,

∠NPD + ∠DPB = 90°

x/2 + ∠DPB = 90°

∠DPB = 90 – x/2

So,

sin∠DPB = sin(90 – x/2) which is not in the choices

However we know that the relationship of sin and cos is:

sin(π/2 - θ) = cos θ

Where,
π/2 = 90
θ = x/2

So,

sin(90 – x/2) = cos(x/2)✓✓✓