# ISEE Upper Level Quantitative : Circles

## Example Questions

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### Example Question #10 : How To Find The Angle Of A Sector

Refer to the above figure, Which is the greater quantity?

(a) The area of

(b) The area of the orange semicircle

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Explanation:

has angles of degree measure 30 and 60; the third angle must measure 90 degrees, making  a right triangle.

For the sake of simplicity, let ; the reasoning is independent of the actual length. The smaller leg of a 30-60-90 triangle has length equal to  times that of the longer leg; this is about

The area of a right triangle is half the product of its legs, so

Also, if , then the orange semicircle has diameter 1 and radius . Its area can be found by substituting  in the formula:

The orange semicircle has a greater area than

### Example Question #191 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

In the above figure,  is a diameter of the circle.

Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) and (b) are equal

Explanation:

That  is a diameter of the circle is actually irrelevant to the problem. Two inscribed angles of a circle that both intercept the same arc, as  and  both do here, have the same measure.

### Example Question #192 : Plane Geometry

is inscribed in a circle.  is a semicircle. .

Which is the greater quantity?

(a)

(b)

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

(a) is the greater quantity

Explanation:

The figure referenced is below:

is a semicircle, so  is one as well; as a semicircle, its measure is . The inscribed angle that intercepts this semicircle, , is a right angle, of measure , and the sum of the measures of the interior angles of a triangle is , so

has greater measure than , so the minor arc intercepted by  , which is , has greater measure than that intercepted by , which is . It follows that the major arc corresponding to the latter, which is , has greater measure than that  corresponding to the former, which is .

### Example Question #24 : Sectors

In the above figure,  is the center of the circle, and . Which is the greater quantity?

(a)

(b)

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

(a) is the greater quantity

Explanation:

Construct . The new figure is below:

, so . It follows that their respective central angles have measures

and

.

Also, since  and  -  being a semicircle - by the Arc Addition Principle, , an inscribed angle which intercepts this arc, has half this measure, which is . The other angle of , which is , also measures , so   is equilateral.

, since all radii are congruent;

by reflexivity;

By the Side-Angle-Side Inequality Theorem (or Hinge Theorem), it follows that . Since  is equilateral, , and since all radii are congruent, . Substituting, it follows that .

### Example Question #194 : Plane Geometry

Trapezoid  is inscribed in a circle, with  a diameter.

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

Explanation:

Below is the inscribed trapezoid referenced, along with its diagonals.

, so, by the Alternate Interior Angles Theorem,

, and their intercepted angles are also congruent - that is,

.

### Example Question #195 : Plane Geometry

In the above figure,  is a diameter of the circle. Which is the greater quantity?

(a)

(b)

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Explanation:

Both  and  are inscribed angles of the same circle which intercept the same arc; they are therefore of the same measure. The fact that  is a diameter of the circle is actually irrelevant to the problem.

### Example Question #196 : Plane Geometry

Figure NOT drawn to scale.

Refer to the above diagram.  is the arithmetic mean of  and .

Which is the greater quantity?

(a)

(b)

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

Explanation:

is the arithmetic mean of  and , so

Also, , or, equivalently,

, so

Solving for :

Also,

If two tangents are drawn to a circle, the measure of the angle they form is half the difference of the measures of the arcs they intercept, so

### Example Question #201 : Geometry

Figure NOT drawn to scale

In the above diagram, .

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Explanation:

is a right triangle whose hypotenuse  has length  times that of leg . This is characteristic of a triangle whose acute angles both have measure  -and consequently, whose acute angles are congruent. Therefore,

These inscribed angles being congruent, the arcs they intercept,  and , are also congruent.

### Example Question #202 : Geometry

Figure NOT drawn to scale

In the above diagram, .

Which is the greater quantity?

(a)

(b)

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Explanation:

is an inscribed angle, so its degree measure is half that of the arc it intercepts, :

.

and  are acute angles of right triangle . They are therefore complimentary - that is, their degree measures total . Consequently,

.

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