### All ISEE Upper Level Quantitative Resources

## Example Questions

### 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

**Possible Answers:**

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

**Correct answer:**

(b) is the greater quantity

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)

**Possible Answers:**

(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

**Correct answer:**

(a) and (b) are equal

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)

**Possible Answers:**

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

**Correct answer:**

(a) is the greater quantity

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)

**Possible Answers:**

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

**Correct answer:**

(a) is the greater quantity

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)

**Possible Answers:**

(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

**Correct answer:**

(a) and (b) are equal

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,

By the Arc Addition Principle,

.

### Example Question #195 : Plane Geometry

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

(a)

(b)

**Possible Answers:**

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

**Correct answer:**

(a) and (b) are equal

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)

**Possible Answers:**

(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

**Correct answer:**

(a) and (b) are equal

is the arithmetic mean of and , so

By arc addition, this becomes

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)

**Possible Answers:**

(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

**Correct answer:**

(a) and (b) are equal

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)

**Possible Answers:**

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

**Correct answer:**

(a) is the greater quantity

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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