### All SAT Math Resources

## Example Questions

### Example Question #31 : Circles

Refer to the above diagram. Evaluate the measure of .

**Possible Answers:**

**Correct answer:**

The total measure of the arcs that comprise a circle is , so from the above diagram,

Substituting the appropriate expression for each arc measure:

Therefore,

and

The measure of the angle formed by the tangent segments and , which is , is half the difference of the measures of the arcs they intercept, so

Substituting:

### Example Question #11 : How To Find The Angle Of A Sector

Figure NOT drawn to scale.

The above figure shows a quadrilateral inscribed in a circle. Evaluate .

**Possible Answers:**

The question cannot be answered from the information given.

**Correct answer:**

If a quadrilateral is inscribed in a circle, then each pair of its opposite angles are supplementary - that is, their degree measures total .

and are two such angles, so

Setting and , and solving for :

,

the correct response.

### Example Question #281 : Plane Geometry

Figure NOT drawn to scale.

The above figure shows a quadrilateral inscribed in a circle. Evaluate .

**Possible Answers:**

The question cannot be answered from the information given.

**Correct answer:**

The question cannot be answered from the information given.

If a quadrilateral is inscribed in a circle, then each pair of its opposite angles are supplementary - that is, their degree measures total .

and are two such angles, so

Setting and , and solving for :

,

The statement turns out to be true regardless of the value of . Therefore, without further information, the value of cannot be determined.

### Example Question #281 : Sat Mathematics

Figure NOT drawn to scale.

The above figure shows a quadrilateral inscribed in a circle. Evaluate .

**Possible Answers:**

**Correct answer:**

If a quadrilateral is inscribed in a circle, then each pair of its opposite angles are supplementary - that is, their degree measures total .

and are two such angles, so

Setting and , and solving for :

,

the correct response.

### Example Question #11 : How To Find The Angle Of A Sector

Figure NOT drawn to scale.

Refer to the above diagram. is a diameter. Evaluate

**Possible Answers:**

**Correct answer:**

is a diameter, so is a semicircle - therefore, . By the Arc Addition Principle,

If we let , then

,

and

If a secant and a tangent are drawn from a point to a circle, the measure of the angle they form is half the difference of the measures of the intercepted arcs. Since and are such segments intercepting and , it holds that

Setting , , and :

The inscribed angle that intercepts this arc, , has half this measure:

.

This is the correct response.

### Example Question #12 : How To Find The Angle Of A Sector

Figure NOT drawn to scale.

In the above figure, is a diameter. Also, the ratio of the length of to that of is 7 to 5. Give the measure of .

**Possible Answers:**

The measure of cannot be determine from the information given.

**Correct answer:**

is a diameter, so is a semicircle, which has measure . By the Arc Addition Principle,

If we let , then, substituting:

,

and

the ratio of the length of to that of is 7 to 5; this is also the ratio of their degree measures; that is,

Setting and :

Cross-multiply, then solve for :

, and

If a secant and a tangent are drawn from a point to a circle, the measure of the angle they form is half the difference of the measures of the intercepted arcs. Since and are such segments whose angle intercepts and , it holds that: