### 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 is equilateral and acute.

The triangle is scalene and right.

The triangle is scalene and acute.

The triangle cannot exist.

The triangle is scalene and obtuse.

**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 #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:**

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:**

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 : Equilateral Triangles

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 : Equilateral Triangles

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 #7 : Equilateral Triangles

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.

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