In circle , points and lie on the circumference. Chords and intersect at point inside the circle. The measure of is , and the measure of minor arc is . What is the measure, in degrees, of minor arc ?

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SAT Circles Question 36 | Free SAT Prep | Aniko

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Question 36·Hard·Circles

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In circle $O$ , points $A, B, C$ , and $D$ lie on the circumference. Chords $\overline{AB}$ and $\overline{CD}$ intersect at point $E$ inside the circle. The measure of $\angle AEC$ is $70^{\circ}$ , and the measure of minor arc $AC$ is $110^{\circ}$ . What is the measure, in degrees, of minor arc $BD$ ?

Your answer

(Express the answer as an integer)

For circle problems with angles formed by intersecting chords, first identify where the angle’s vertex is: center (central), on the circle (inscribed), or inside/outside the circle (intersecting chords or secants). For chords intersecting inside the circle, remember the key rule: the angle equals half the sum of the intercepted arcs. Quickly label the relevant arcs, set up a simple linear equation, and solve; this is usually faster and less error-prone than trying to reason purely from a mental picture. Double-check that you didn’t accidentally use the “half the difference” formula, which applies only when the vertex is outside the circle.

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Identify the type of angle

Is $\angle AEC$ an inscribed angle (with vertex on the circle), a central angle (with vertex at the center), or an angle formed by intersecting chords inside the circle? The formula you use depends on this.

Think about which arcs the angle intercepts

The sides of $\angle AEC$ go through points $A$ and $C$ . Its vertical angle involves points $B$ and $D$ . Together, these angles intercept two arcs on the circle — which arcs have endpoints at these four points?

Use the correct formula for intersecting chords

For angles formed by intersecting chords inside a circle, the angle equals half the sum of the intercepted arcs, not the difference. Write an equation using $70^{\circ}$ , $110^{\circ}$ , and the unknown arc $BD$ , then solve for that unknown.

Desmos Guide

Model the relationship with two graphs

In Desmos, type y = (110 + x)/2 on one line and y = 70 on another line. Then find the point where the two graphs intersect; the x-coordinate of that intersection is the measure of minor arc $BD$ in degrees.

Step-by-step Explanation

Recall the intersecting chords angle–arc relationship

When two chords intersect inside a circle, the measure of the angle formed is half the sum of the measures of the arcs intercepted by that angle and its vertical angle.

Here, chords $\overline{AB}$ and $\overline{CD}$ intersect at $E$ , forming $\angle AEC$ and its vertical angle $\angle BED$ .

$\angle AEC$ and $\angle BED$ together intercept minor arc $AC$ and minor arc $BD$ .

So the relationship is:

m\angle AEC = \frac{1}{2}\big(m\widehat{AC} + m\widehat{BD}\big)

Substitute the known measures into the formula

You are given:

$m\angle AEC = 70^{\circ}$
$m\widehat{AC} = 110^{\circ}$

Let $x$ be the measure (in degrees) of minor arc $BD$ .

Substitute into the formula:

70 = \frac{1}{2}(110 + x)

This equation relates the known angle and arc to the unknown arc $x$ .

Solve the equation for arc BD

Solve the equation

70 = \frac{1}{2}(110 + x)

Step 1: Multiply both sides by $2$ :

140 = 110 + x

Step 2: Subtract $110$ from both sides:

30 = x

So the measure of minor arc $BD$ is $\boxed{30}$ degrees.

Strategy

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation