We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : Geometry

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

Create An Account

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Geometry

Obtuse

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

(a) x+ y

(b)

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

Extend $\overline{AB}$ as seen in the figure below:

Obtuse

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles; specifically,

$m \angle CAD = m \angle B + m \angle C$ ,

and

$m \angle CAD =( x+ y) ^{\circ }$

However, $m \angle CAD > z ^{\circ }$ , so, by substitution,

x+y>z

Report an Error

Example Question #11 : Plane Geometry

Given: $\bigtriangleup ABC$ . AB= AC = 10, BC = 12 . Which is the greater quantity?

(a) $m \angle B$

(b) $60^{\circ }$

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

Below is the referenced triangle along with $\bigtriangleup DEF$ , an equilateral triangle with sides of length 10:

Triangles

As an angle of an equilateral triangle, $\angle D$ has measure $60 ^{\circ }$ . Applying the Side-Side-Side Inequality Theorem, since AB = DE , AC = DF , and BC > EF , it follows that $m \angle A > m \angle D$ , so $m \angle A > 60^{\circ }$ .

Also, since AB = BC , by the Isosceles Triangle Theorem, $m \angle B = m \angle C$ . Since $m \angle A > 60^{\circ }$ , and the sum of the measures of the angles of a triangle is $180 ^{\circ }$ , it follows that

$m \angle B + m \angle C < 120^{\circ }$

Substituting and solving:

$m \angle B + m \angle B < 120^{\circ }$

$2 m \angle B < 120^{\circ }$

$2 m \angle B \div 2 < 120^{\circ } \div 2$

$m \angle B < 60^{\circ }$ .

Report an Error

Example Question #11 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Given $\Delta ABC$ and $\Delta DEF$ with $AC = DF, AB = DE, m \angle B = m \angle E$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Examine the diagram below, in which two triangles matching the given descriptions have been superimposed.

Ssa

Note that AB = DE and $m \angle B = m \angle E$ . The two question marks need to be replaced by and . No matter how you place these two points, AC = DF . However, with one replacement, BC > EF ; with the other replacement, BC < EF . Therefore, the information given is insufficient to answer the question.

Report an Error

Example Question #12 : Plane Geometry

Consider $\Delta ABC$ and $\Delta DEF$ with AC = DF, AB = DE, BC = EF .

Which is the greater quantity?

(a) $m \angle A$

(b) $m \angle D$

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

AC = DF, AB = DE, BC = EF , so, by the Side-Side-Side Principle, since there are three pairs of congruent corresponding sides between the triangles, we can say they are congruent - that is,

$\Delta ABC \cong \Delta DEF$ .

Corresponding angles of congruent sides are congruent, so $m \angle A = m \angle D$ .

Report an Error

Example Question #1 : How To Find The Length Of The Side Of A Triangle

Which of the following could be the lengths of the three sides of a scalene triangle?

Possible Answers:

All of the other choices are possible lengths of a scalene triangle

$5,000 \textrm{ m, } 4,000 \textrm{ m, }8,000 \textrm{ m}$

$6 \textrm{ in, } 8 \textrm{ in, }10 \textrm{ in}$

$0.7 \textrm{ in, } 0.9 \textrm{ in, }1.4 \textrm{ in}$

$8 \textrm{ cm, } 9 \textrm{ cm, }10 \textrm{ cm}$

Correct answer:

All of the other choices are possible lengths of a scalene triangle

Explanation:

A scalene triangle, by definition, has sides all of different lengths. Since all of the given choices fit that criterion, the correct choice is that all can be scalene.

Report an Error

Example Question #12 : Geometry

Given $\Delta ABC$ with right angle $\angle B$ , $m \angle C = 50 ^{\circ }$ .

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

$m \angle B = 90^{\circ}$

$m \angle C = 50 ^{\circ }$

The sum of the measures of the angles of a triangle is 180, so

$m \angle A + m \angle B + m \angle C = 180$

$m \angle A +90 + 50 = 180$

$m \angle A +140= 180$

$m \angle A +140-140= 180-140$

$m \angle A = 40^{\circ }$

$m \angle C > m \angle A$ , so the side opposite $\angle C$ , which is $\overline{AB}$ , is longer than the side opposite $\angle A$ , which is $\overline{BC}$ . This makes (a) the greater quantity.

Report an Error

Example Question #3 : How To Find The Length Of The Side Of A Triangle

Given $\Delta ABC$ with obtuse angle $\angle A$ , which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

To compare the lengths of $\overline{AB}$ and $\overline{BC}$ from the angle measures, it is necessary to know which of their opposite angles - $\angleC$ $\angle C$ and $\angle A$ , respectively - is the greater angle. Since $\angle A$ is the obtuse angle, it has the greater measure, and $\overline{BC}$ is the longer side. This makes (b) greater.

Report an Error

Example Question #1 : How To Find The Length Of The Side Of A Triangle

$\Delta ABC$ has obtuse angle $\angle B$ ; AB = 7, BC = 24 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

Since $\angle B$ is the obtuse angle of $\Delta ABC$ ,

$\left ( AC \right )^{2} >\left ( AB \right )^{2} + \left ( BC \right )^{2} = 7^{2} + 24^{2} = 49 + 576 = 625$ .

$\left ( AC \right )^{2} > 625$ ,

$AC > \sqrt{625}$

AC >25 ,

so (a) is the greater quantity.

Report an Error

Example Question #2 : How To Find The Length Of The Side Of A Triangle

Given $\Delta ABC$ with AB = 5, BC = 12 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

Use the Triangle Inequality:

AC < AB + BC

AC < 5 + 12

AC < 17

This makes (b) the greater quantity.

Report an Error

Example Question #11 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Given $\Delta ABC$ with AB = 5, BC = 12 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

By the Converse of the Pythagorean Theorem,

$AC = \sqrt{(AB)^{2}+(BC)^{2}}= \sqrt{5^{2}+12^{2}}= \sqrt{25+144}=\sqrt{169} = 13$

if and only if $\angle B$ is a right angle.

However, if $\angle B$ is acute, then AC < 13 ; if $\angle B$ is obtuse, then AC > 13 .

Since we do not know whether $\angle B$ is acute, right, or obtuse, we cannot determine whether (a) or (b) is greater.

Report an Error

View Tutors

Gabby
Certified Tutor

University of Oregon, Bachelor in Arts, Journalism. University of Oregon, Current Grad Student, Communication, General.

View Tutors

Ana
Certified Tutor

Grand Canyon University, Bachelors, Education. Grand Canyon University, Masters, Education.

View Tutors

Crystal
Certified Tutor

SUNY at Binghamton, Bachelor of Science, Mechanical Engineering.

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in Denver, English Tutors in Washington DC, English Tutors in Boston, Chemistry Tutors in Phoenix, Algebra Tutors in Washington DC, Calculus Tutors in Houston, Physics Tutors in Washington DC, Algebra Tutors in Phoenix, Statistics Tutors in Miami, GRE Tutors in Philadelphia

Popular Courses & Classes

GMAT Courses & Classes in Washington DC, GMAT Courses & Classes in Boston, MCAT Courses & Classes in San Diego, ISEE Courses & Classes in Boston, ACT Courses & Classes in Atlanta, ISEE Courses & Classes in Philadelphia, SSAT Courses & Classes in Denver, MCAT Courses & Classes in Boston, ACT Courses & Classes in Philadelphia, GRE Courses & Classes in Phoenix

Popular Test Prep

SSAT Test Prep in New York City, LSAT Test Prep in Boston, ACT Test Prep in Atlanta, ISEE Test Prep in Seattle, LSAT Test Prep in Atlanta, ACT Test Prep in Dallas Fort Worth, LSAT Test Prep in Philadelphia, GMAT Test Prep in Washington DC, SSAT Test Prep in Washington DC, ACT Test Prep in Philadelphia

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

ISEE Upper Level Quantitative : Geometry

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #11 : Geometry

Example Question #11 : Plane Geometry

Example Question #11 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #12 : Plane Geometry

Example Question #1 : How To Find The Length Of The Side Of A Triangle

Example Question #12 : Geometry

Example Question #3 : How To Find The Length Of The Side Of A Triangle

Example Question #1 : How To Find The Length Of The Side Of A Triangle

Example Question #2 : How To Find The Length Of The Side Of A Triangle

Example Question #11 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: