PSAT Math : Equilateral Triangles

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 : Equilateral Triangles

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

Possible Answers:

The triangle is scalene and obtuse.

The triangle is equilateral and acute.

The triangle is scalene and right.

The triangle cannot exist.

The triangle is scalene and acute.

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 : 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:

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 : 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:

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 #2 : 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 #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  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 #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:

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 #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:

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.

Example Question #2 : Equilateral Triangles

The area of square ABCD is 50% greater than the perimeter of the equilateral triangle EFG. If the area of square ABCD is equal to 45, then what is the area of EFG?

Possible Answers:

30

50√3

25

25√3

50

Correct answer:

25√3

Explanation:

If the area of ABCD is equal to 45, then the perimeter of EFG is equal to x * 1.5 = 45. 45 / 1.5 = 30, so the perimeter of EFG is equal to 30. This means that each side is equal to 10.

The height of the equilateral triangle EFG creates two 30-60-90 triangles, each with a hypotenuse of 10 and a short side equal to 5. We know that the long side of 30-60-90 triangle (here the height of EFG) is equal to √3 times the short side, or 5√3.

We then apply the formula for the area of a triangle, which is 1/2 * b * h. We get 1/2 * 10 * 5√3 = 5 * 5√3 = 25√3.

In general, the height of an equilateral triangle is equal to √3 / 2 times a side of the equilateral triangle. The area of an equilateral triangle is equal to 1/2 * √3s/ 2 * s = √3s2/4.

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

What is the area of an equilateral triangle with sides 12 cm?

Possible Answers:

12√2

36√3

18√3

54√2

72√3

Correct answer:

36√3

Explanation:

An equilateral triangle has three congruent sides and results in three congruent angles. This figure results in two special right triangles back to back: 30° – 60° – 90° giving sides of x - x √3 – 2x in general. The height of the triangle is the x √3 side.  So Atriangle = 1/2 bh = 1/2 * 12 * 6√3 = 36√3 cm2.

Example Question #1 : Equilateral Triangles

An equilateral triangle has a perimeter of 18. What is its area?

Possible Answers:

Correct answer:

Explanation:

Recall that an equilateral triangle also obeys the rules of isosceles triangles.   That means that our triangle can be represented as having a height that bisects both the opposite side and the angle from which the height is "dropped."  For our triangle, this can be represented as:

6-equilateral

Now, although we do not yet know the height, we do know from our 30-60-90 regular triangle that the side opposite the 60° angle is √3 times the length of the side across from the 30° angle. Therefore, we know that the height is 3√3.

Now, the area of a triangle is (1/2)bh. If the height is 3√3 and the base is 6, then the area is (1/2) * 6 * 3√3 = 3 * 3√3 = 9√(3).

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