ISEE Upper Level Quantitative : How to find the angle of a sector

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

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Example Question #191 : Plane Geometry

Inscribed angle 2

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) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

(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) 

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

Explanation:

The figure referenced is below:

Inscribed angle

 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

Inscribed angle 3

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

Explanation:

Construct . The new figure is below:

 Inscribed angle 3

, 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

Explanation:

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

Inscribed angle 3

, 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

Circle 2

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

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

Tangents 1

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

Explanation:

 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

Inscribed angle 4

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

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

Inscribed angle 4

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

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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