In the circle shown, chords and intersect at point . Angle measures , and angle measures . If the circumference of the circle is 180 inches, what is the length of minor arc ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 44 | Free SAT Prep | Aniko

00:00

Question 44·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

0 / 200

In the circle shown, chords $AC$ and $BD$ intersect at point $E$ . Angle $AEB$ measures $73^\circ$ , and angle $CBD$ measures $31^\circ$ . If the circumference of the circle is 180 inches, what is the length of minor arc $AB$ ?

For circle problems, first classify each given angle by where its vertex is: on the circle (inscribed angle) or inside the circle where chords intersect (intersecting-chords angle). Convert angles to arc measures using the correct rule, solve for the needed arc in degrees, and only then convert degrees to length using $\frac{\text{arc degrees}}{360}\times \text{circumference}$ .

Hints

Click me!

Start with the inscribed angle

Angle $CBD$ has its vertex on the circle. Which arc does it intercept, and how does an inscribed angle relate to its intercepted arc?

Use the intersecting chords fact

Angle $AEB$ is formed by two chords crossing inside the circle. Its measure is half the sum of two intercepted arcs.

Turn degrees into length

Once you have $m(\widehat{AB})$ in degrees, take that fraction of the full circumference (out of $360^\circ$ ).

Desmos Guide

Compute the intercepted arc from the inscribed angle

In Desmos, define arcCD = 2*31 to represent $m(\widehat{CD})$ in degrees.

Use the intersecting-chords relationship to get the arc measure

Define arcAB = 2*73 - arcCD to represent $m(\widehat{AB})$ in degrees.

Convert arc measure to arc length

Define L = (arcAB/360)*180. The value of L is the length of minor arc $AB$ in inches.

Step-by-step Explanation

Use the inscribed angle to find an arc measure

Angle $CBD$ is an inscribed angle, so it intercepts arc $CD$ .

For an inscribed angle,

\text{(inscribed angle)}=\frac{1}{2}(\text{intercepted arc})

So,

m(\widehat{CD})=2\cdot 31^\circ=62^\circ

Relate arcs using intersecting chords

When two chords intersect inside a circle, the measure of the angle formed equals half the sum of the measures of the intercepted arcs.

Angle $AEB$ intercepts arcs $AB$ and $CD$ , so

73^\circ=\frac{1}{2}\big(m(\widehat{AB})+m(\widehat{CD})\big)

Substitute $m(\widehat{CD})=62^\circ$ :

73=\frac{1}{2}(m(\widehat{AB})+62)

Multiply by 2:

146=m(\widehat{AB})+62

So,

m(\widehat{AB})=84^\circ

Convert arc measure to arc length

Arc length is the same fraction of the circumference as its degree measure is of $360^\circ$ :

\text{arc length}= \frac{84}{360}\cdot 180

Simplify:

\frac{84}{360}=\frac{7}{30} \quad\Rightarrow\quad \frac{7}{30}\cdot 180=7\cdot 6=42

Therefore, the length of minor arc $AB$ is 42 inches.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation