We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

PSAT Math : How to find the length of the side of an equilateral triangle

Study concepts, example questions & explanations for PSAT Math

Create An Account

All PSAT Math Resources

10 Diagnostic Tests 421 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

Which of the following describes a triangle with sides of length one meter, 100 centimeters, and 10 decimeters?

Possible Answers:

The triangle cannot exist.

The triangle is scalene and right.

The triangle is equilateral and acute.

The triangle is scalene and acute.

The triangle is scalene and obtuse.

Correct answer:

The triangle is equilateral and acute.

Explanation:

One meter, 100 centimeters, and 10 decimeters are all equal to the same quantity. This makes the triangle equilateral and, subsequently, acute.

Report an Error

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

Two triangles have the same area. One is an equilateral triangle. The other is a right triangle with hypotenuse 12 and one leg of length 8. Give the sidelength of the equilateral triangle to the nearest tenth.

Possible Answers:

9.1

12.9

10.1

27.2

10.5

Correct answer:

9.1

Explanation:

A right triangle with hypotenuse 12 and leg 8 also has leg

$a = \sqrt{12^{2}-8^{2}} = \sqrt{144-64} = \sqrt{80} \approx 8.944$

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

$A \approx \frac{1}{2} \cdot 8 \cdot 8.944 \approx 35.777$ ,

which is also the area of the given equilateral triangle.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

so if we set $A \approx 35.777$ , we can solve for :

$\frac{s^{2}\sqrt{3}}{4} \approx35.777$

$s^{2} \approx35.777 \cdot \frac{4}{\sqrt{3}} \approx 35.777 \cdot \frac{4}{1.732} \approx82.623$

$s \approx \sqrt{82.623} \approx 9.090$

The correct choice is 9.1.

Report an Error

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

An equilateral triangle has the same area as a circle with circumference 100. To the nearest tenth, give the sidelength of the triangle.

Possible Answers:

42.9

47.4

67.1

60.6

269.4

Correct answer:

42.9

Explanation:

The circle with circumference 100 has radius

$r = \frac{C}{2 \pi} = \frac{100 }{2 \pi} = \frac{50}{\pi}$

Its area is

$A = \pi r^{2}$

$A = \pi \cdot \left ( \frac{50}{\pi} \right )^{2}$

$A = \pi \cdot \frac{2,500}{\pi ^{2}}$

$A = \frac{2,500}{\pi }$

We can substitute this for in the equation for the area of an equilateral triangle, and solve for :

$\frac{s^{2}\sqrt{3}}{4} =A$

$\frac{s^{2}\sqrt{3}}{4} = \frac{2,500}{\pi}$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}} = \frac{2,500}{\pi} \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{2,500}{\pi} \cdot \frac{4}{\sqrt{3}} \approx \frac{10,000}{3.14159\cdot 1.732} \approx 1,837.76$

$s \approx \sqrt{1,837.76} \approx 42.9$

The correct response is 42.9.

Report an Error

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

Two triangles have the same area. One is an equilateral triangle. The other is an isosceles right triangle with hypotenuse . Give the sidelength of the equilateral triangle in terms of .

Possible Answers:

$\frac{ x} { \sqrt{2}}$

$\frac{ x} { \sqrt[4]{3}}$

$\frac{x}{3}$

$\frac{ x} { \sqrt{3}}$

$\frac{ x} { \sqrt[4]{2}}$

Correct answer:

$\frac{ x} { \sqrt[4]{3}}$

Explanation:

An isosceles right triangle is also a $45^{\circ } - 45^{\circ } - 90^{\circ }$ triangle, whose legs each measure the length of the hypotenuse divided by $\sqrt{2}$ . Therefore, since the hypotenuse measures , each leg measures $\frac{x}{\sqrt{2}}$ .

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

$A = \frac{1}{2} \cdot \frac{x}{\sqrt{2}} \cdot \frac{x}{\sqrt{2}} = \frac{x^{2}}{4}$

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$ ,

so set $A = \frac{x^{2}}{4}$ and solve for :

$\frac{s^{2}\sqrt{3}}{4} = \frac{x^{2}}{4}$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4} { \sqrt{3}}= \frac{x^{2}}{4} \cdot \frac{4} { \sqrt{3}}$

$s^{2} = \frac{x^{2}} { \sqrt{3}}$

$s =\sqrt{ \frac{x^{2}} { \sqrt{3}}}$

$s =\frac{ x} { \sqrt[4]{3}}$

Report an Error

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

Two triangles have the same area. One is an equilateral triangle. The other is a $30^{\circ }- 60^{\circ }-90^{\circ }$ right triangle with hypotenuse . Give the sidelength of the equilateral triangle in terms of .

Possible Answers:

$\frac{x }{3}$

$\frac{x }{2}$

$\frac{x\sqrt{2} }{2}$

$\frac{x\sqrt{6} }{2}$

$\frac{x\sqrt{3} }{3}$

Correct answer:

$\frac{x\sqrt{2} }{2}$

Explanation:

A $30^{\circ }- 60^{\circ }-90^{\circ }$ right triangle has a short leg half as long as its hypotenuse , which is $\frac{x}{2}$ . Its long leg is $\sqrt{3}$ times as long as its short leg, which will be $\frac{x\sqrt{3}}{2}$ . Its area is half the product of its legs, so the area will be

$A = \frac{1}{2} \cdot \frac{x}{2} \cdot \frac{x\sqrt{3}}{2} = \frac{x^{2}\sqrt{3}}{8}$

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$ ,

so set $A = \frac{x^{2}\sqrt{3}}{8}$ and solve for :

$\frac{s^{2}\sqrt{3}}{4} = \frac{x^{2}\sqrt{3}}{8}$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4} {\sqrt{3}} = \frac{x^{2}\sqrt{3}}{8} \cdot \frac{4} {\sqrt{3}}$

$s^{2} = \frac{x^{2} }{2}$

$s =\sqrt{ \frac{x^{2} }{2}}$

$s = \frac{x }{\sqrt{2}} = \frac{x\sqrt{2}}{2}$

Report an Error

Example Question #5 : How To Find The Length Of The Side Of An Equilateral Triangle

A square and an equilateral triangle have the same area. Call the side length of the square . Give the side length of the equilateral triangle in terms of .

Possible Answers:

$\frac{ 4x}{ \sqrt{3}}$

$\frac{ 2x}{ \sqrt{3}}$

$2x \sqrt{3}$

$\frac{ 4x}{ \sqrt[4]{3}}$

$\frac{ 2x}{ \sqrt[4]{3}}$

Correct answer:

$\frac{ 2x}{ \sqrt[4]{3}}$

Explanation:

The area of a square is A=s^2 where represents the side length. In our case the side length is therefore, the area of the square is $x^{2}$ ; this will also be the area of the equilateral triangle.

The formula for the area of an equilateral triangle with sidelength is

$A = \frac{s^{2} \sqrt{3}}{4}$

If we let $A = x^{2}$ , we can solve for in the equation:

$\frac{s^{2} \sqrt{3}}{4} = x^{2}$

$s^{2} =\frac{ 4 x^{2}}{ \sqrt{3}}$

$s =\sqrt{\frac{ 4 x^{2}}{ \sqrt{3}}}$

$s = \frac{ \sqrt{4 x^{2}}}{ \sqrt{\sqrt{3}}}$

$s =\frac{ 2x}{ \sqrt[4]{3}}$

which is the correct response.

Report an Error

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

A regular hexagon and an equilateral triangle have the same area. Call the side length of the hexagon . Give the side length of the equilateral triangle in terms of .

Possible Answers:

$x\sqrt{6}$

$x\sqrt{3}$

Correct answer:

$x\sqrt{6}$

Explanation:

A regular hexagon can be divided by its three diameters into six congruent equilateral triangles. Since each triangle will have sidelength , each will have area equal to

$A' = \frac{x^{2}\sqrt{3}}{4}$

Multiply by 6 to get the area of the hexagon:

$A = 6A' = 6 \cdot \frac{x^{2}\sqrt{3}}{4} = \frac{3x^{2}\sqrt{3}}{2}$

We can substitute this for in the equation for the area of an equilateral triangle, and solve for :

$\frac{s^{2}\sqrt{3}}{4} =A$

$\frac{s^{2}\sqrt{3}}{4} =\frac{3x^{2}\sqrt{3}}{2}$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}} =\frac{3x^{2}\sqrt{3}}{2} \cdot \frac{4}{\sqrt{3}}$

$s^{2}= 6x^{2}$

$s = \sqrt{6x^{2}} = x\sqrt{6}$ , the correct response.

Report an Error

View PSAT Mathematics Tutors

Crystal
Certified Tutor

SUNY at Binghamton, Bachelor of Science, Mechanical Engineering.

View PSAT Mathematics Tutors

Dr. Derrick
Certified Tutor

Huston-Tillotson University, Bachelor of Science, Mathematics. Grand Canyon University, Master of Arts, Education.

View PSAT Mathematics Tutors

Eva
Certified Tutor

Technicka Univerzita Liberec, Bachelor of Education, Education.

All PSAT Math Resources

10 Diagnostic Tests 421 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Chicago, Statistics Tutors in Boston, Math Tutors in Denver, Calculus Tutors in Boston, French Tutors in Atlanta, Biology Tutors in New York City, SAT Tutors in Houston, Algebra Tutors in Miami, Reading Tutors in Los Angeles, LSAT Tutors in Boston

Popular Courses & Classes

GRE Courses & Classes in New York City, MCAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Miami, Spanish Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in New York City, Spanish Courses & Classes in Denver, ISEE Courses & Classes in Philadelphia, GRE Courses & Classes in Boston, ISEE Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Dallas Fort Worth

Popular Test Prep

ISEE Test Prep in New York City, MCAT Test Prep in Philadelphia, GMAT Test Prep in New York City, MCAT Test Prep in Boston, SAT Test Prep in Miami, SAT Test Prep in Denver, ISEE Test Prep in Washington DC, MCAT Test Prep in New York City, MCAT Test Prep in Chicago, MCAT Test Prep in Dallas Fort Worth

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

PSAT Math : How to find the length of the side of an equilateral triangle

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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

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

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

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

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

Example Question #5 : How To Find The Length Of The Side Of An Equilateral Triangle

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

All PSAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: