Question

The figure shows a circle with center

P
, a diameter
¯¯¯¯¯¯
B
D
, and inscribed
△
B
C
D
.
P
C

=

10
.
Let
m
∠
C
B
D
=
(
x
)
°
and
m
∠
B
C
D
=
(
x
+
54
)
°
.

Answer 1

From the given information, we have:

PC = 10 (length of segment PC)

∠CBD = x°

∠BCD = x + 54°

We can determine the relationship between angles ∠CBD and ∠BCD by recognizing that they are inscribed angles intercepting the same arc CD. In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. Therefore, we have:

∠BCD = 1/2(arc CD)

Since ∠BCD = x + 54° and arc CD is the diameter, which is 180°, we can set up the following equation:

x + 54 = 1/2(180)

Simplifying the equation:

x + 54 = 90

x = 90 - 54

x = 36

So, the measure of ∠CBD (m∠CBD) is 36°.