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

Study concepts, example questions & explanations for PSAT Math

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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 is equilateral and acute.

The triangle is scalene and obtuse.

The triangle cannot exist.

The triangle is scalene and acute.

The triangle is scalene and right.

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.

Example Question #2 : Equilateral Triangles

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:

Correct answer:

Explanation:

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

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

,

which is also the area of the given equilateral triangle.

The area of an equilateral triangle is given by the formula

so if we set , we can solve for :

The correct choice is 9.1.

Example Question #3 : Equilateral Triangles

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:

Correct answer:

Explanation:

The circle with circumference 100 has radius

Its area is 

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

The correct response is 42.9.

Example Question #4 : Equilateral Triangles

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:

Correct answer:

Explanation:

An isosceles right triangle is also a  triangle, whose legs each measure the length of the hypotenuse divided by . Therefore, since the hypotenuse measures , each leg measures 

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

The area of an equilateral triangle is given by the formula

,

so set  and solve for :

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 a  right triangle with hypotenuse .  Give the sidelength of the equilateral triangle in terms of .

Possible Answers:

Correct answer:

Explanation:

 right triangle has a short leg half as long as its hypotenuse , which is . Its long leg is  times as long as its short leg, which will be . Its area is half the product of its legs, so the area will be

The area of an equilateral triangle is given by the formula

,

so set  and solve for :

Example Question #2 : 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:

Correct answer:

Explanation:

The area of a square is  where  represents the side length. In our case the side length is  therefore, the area of the square is ; this will also be the area of the equilateral triangle.

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

If we let , we can solve for  in the equation:

which is the correct response.

Example Question #4 : 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:

Correct answer:

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 

Multiply by 6 to get the area of the hexagon:

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

, the correct response.

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