In the circle shown, points , , , and lie on the circle in that order. Chords and intersect at point . The measure of is degrees. The measures of arcs , , and are degrees, degrees, and degrees, respectively. Which choice is the value of ? Note: Figure not drawn to scale.

Solution available with step-by-step explanation

Super-Hard SAT Math Question 189 | Free SAT Prep | Aniko

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Question 189·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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In the circle shown, points $A$ , $B$ , $C$ , and $D$ lie on the circle in that order. Chords $AC$ and $BD$ intersect at point $E$ .

The measure of $\angle AEB$ is $2x+79$ degrees. The measures of arcs $AB$ , $BC$ , and $DA$ are $5x+10$ degrees, $3x+20$ degrees, and $56$ degrees, respectively.

Which choice is the value of $x$ ?

Note: Figure not drawn to scale. Аniкo ‍-‌ Fr‌ee ЅАT ‍Рrер

First translate the circle information into arc measures: since the points are in order, the four arcs around the circle sum to $360^\circ$ , letting you write arc $CD$ in terms of $x$ . Then apply the key theorem for this setup: an angle formed by two intersecting chords inside a circle equals half the sum of the opposite intercepted arcs (here, $AB$ and $CD$ ). Clear the fraction by multiplying by $2$ , solve the linear equation, and match your result to the choices. (An‍іко.аі⁠)

Hints

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Find the arc that is not given

Use that the four arcs $AB$ , $BC$ , $CD$ , and $DA$ go all the way around the circle, so their measures add to $360^\circ$ .

Recall the chord-chord angle rule

For two chords intersecting inside a circle, the angle formed equals half the sum of the measures of the arcs intercepted by that angle and its vertical angle. Сo‍ntеnt b‍у Aniko.аі

Keep the fraction until the end

After you substitute, multiply both sides of the equation by $2$ to clear the fraction before solving for $x$ .

Desmos Guide

Enter the angle expression

In Desmos, enter

$y_1=2x+79$

Enter the arc-sum expression for arc $CD$ inside the formula

Enter

y_2=\frac{(5x+10)+(360-(5x+10)-(3x+20)-56)}{2}

This represents $\tfrac{1}{2}(\text{arc }AB+\text{arc }CD)$ , with arc $CD$ found from the $360^\circ$ total.

Find the intersection

Click the intersection point of $y_1$ and $y_2$ . The $x$ -coordinate of that intersection is the value of $x$ . © Anі‌кo

Step-by-step Explanation

Find the missing arc $CD$

Since points $A$ , $B$ , $C$ , and $D$ go around the circle in order,

\text{arc }AB+\text{arc }BC+\text{arc }CD+\text{arc }DA=360

So,

\begin{aligned} \text{arc }CD &=360-(5x+10)-(3x+20)-56 \\ &=360-(8x+86) \\ &=274-8x \end{aligned}

Use the intersecting chords angle theorem

When two chords intersect inside a circle,

m\angle AEB=\frac{1}{2}\big(\text{arc }AB+\text{arc }CD\big)

Substitute the given expressions:

2x+79=\frac{1}{2}\big((5x+10)+(274-8x)\big)

Solve for $x$

Simplify the right-hand side:

\frac{1}{2}\big((5x+10)+(274-8x)\big)=\frac{1}{2}(284-3x)

So,

2x+79=\frac{284-3x}{2}

Multiply both sides by $2$ :

4x+158=284-3x

Add $3x$ to both sides and subtract $158$ from both sides:

7x=126

Select the correct answer choice

Divide by $7$ : А‌nікο Queѕtiο⁠n B‌ank

x=18

So the correct choice is $18$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation