### All PSAT Math Resources

## 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 cannot exist.

The triangle is equilateral and acute.

The triangle is scalene and obtuse.

The triangle is scalene and acute.

The triangle is scalene and right.

**Correct answer:**

The triangle is equilateral and acute.

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

### 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:**

**Correct answer:**

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 #1 : 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:**

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 : 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:**

**Correct answer:**

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 #5 : 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:**

A 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 #6 : 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:**

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 #541 : Psat Mathematics

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:**

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.