Question

18 if the length of \\( \overline{ab} \\) is 13 millimeters, what is the length of \\( \overline{cd} \\)?
19 if the measure of \\( \angle aob = 105^\circ \\), what is the measure of \\( \angle doc \\)?
20 use the diagram to the right to answer the following questions
a if \\( m\overline{ae} = 10 \text{cm} \\) and \\( m\overline{ec} = 3x - 8 \\), what is the value of \\( x \\)?
b if \\( m\overline{ab} = 12y - 7 \\) and \\( m\overline{bc} = 5y + 21 \\), what is \\( m\overline{bc} \\)?

Explanation:

20a Solution:

Step1: Identify equal segments

Since \( AB \) and \( CD \) are diameters, and \( OE \perp AC \), \( AE = EC \) (perpendicular from center to chord bisects the chord). Given \( m\overline{AE}=10 \, \text{cm} \), so \( m\overline{EC}=10 \, \text{cm} \).

Step2: Set up equation

We know \( m\overline{EC}=3x - 8 \), and \( m\overline{EC}=10 \). So:
\( 3x - 8 = 10 \)

Step3: Solve for \( x \)

Add 8 to both sides: \( 3x = 10 + 8 = 18 \)
Divide by 3: \( x=\frac{18}{3}=6 \)

Step1: Identify equal arcs

Since \( AB \) and \( CD \) are perpendicular diameters, \( \overset{\frown}{AB}=\overset{\frown}{BC}=\overset{\frown}{CD}=\overset{\frown}{DA} \)? Wait, no, actually, \( AB \) and \( CD \) are perpendicular, so \( \angle AOB=\angle BOC = 90^\circ \)? Wait, no, the diagram shows \( AB \) and \( CD \) as perpendicular diameters, so arcs \( AB \) and \( BC \)? Wait, no, the problem says \( m\overset{\frown}{AB}=12y - 7 \) and \( m\overset{\frown}{BC}=5y + 21 \). Wait, actually, since \( AB \) and \( CD \) are perpendicular diameters, \( \overset{\frown}{AB}=\overset{\frown}{BC} \)? Wait, no, maybe \( AB \) and \( BC \) are arcs such that \( AB = BC \) (if the diameters are perpendicular, each arc is \( 90^\circ \), but here it's length? Wait, no, maybe the arcs are equal. Wait, the problem might have \( AB \) and \( BC \) as arcs with equal measure (since \( OE \perp AC \) and \( CD \) is diameter, maybe \( AB = BC \)). So set \( 12y - 7 = 5y + 21 \)

Step2: Solve for \( y \)

Subtract \( 5y \) from both sides: \( 12y - 5y - 7 = 21 \)
\( 7y - 7 = 21 \)
Add 7: \( 7y = 21 + 7 = 28 \)
Divide by 7: \( y = 4 \)

Step3: Find \( m\overset{\frown}{BC} \)

Substitute \( y = 4 \) into \( 5y + 21 \):
\( 5(4)+21 = 20 + 21 = 41 \)
Wait, that can't be, because if they are perpendicular diameters, arcs should be \( 90^\circ \). Wait, maybe the problem is that \( AB \) and \( BC \) are arcs such that \( AB = BC \) (since the diameters are perpendicular, so \( \angle AOB=\angle BOC = 90^\circ \), so arcs \( AB \) and \( BC \) have equal measure). So the equation is \( 12y - 7 = 5y + 21 \)
Wait, solving:
\( 12y - 5y = 21 + 7 \)
\( 7y = 28 \)
\( y = 4 \)
Then \( m\overset{\frown}{BC}=5(4)+21 = 20 + 21 = 41 \). But that contradicts the perpendicular diameter. Wait, maybe the diagram is different. Alternatively, maybe \( AB \) and \( BC \) are chords, but the problem says \( m\overset{\frown}{AB} \) and \( m\overset{\frown}{BC} \). Anyway, following the problem's equation:

Step1: Set arcs equal (assuming \( \overset{\frown}{AB}=\overset{\frown}{BC} \))

\( 12y - 7 = 5y + 21 \)

Step2: Solve for \( y \)

\( 12y - 5y = 21 + 7 \)
\( 7y = 28 \)
\( y = 4 \)

Step3: Calculate \( m\overset{\frown}{BC} \)

Substitute \( y = 4 \) into \( 5y + 21 \):
\( 5(4)+21 = 20 + 21 = 41 \)

Answer:

\( x = 6 \)

20b Solution: