PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 : Triangles

Which of the following sets of line-segment lengths can form a triangle?

Possible Answers:

5,12,17

2,3,5

5,12,13

7,9,16

10,15,25

Correct answer:

5,12,13

Explanation:

In any given triangle, the sum of any two sides is greater than the third.  The incorrect answers have the sum of two sides equal to the third.

Example Question #462 : Plane Geometry

In right \Delta ABC, \angle ABC = 2x and \angle BCA= \frac{x}{2}.

What is the value of x?

Possible Answers:

48

36

30

24

32

Correct answer:

36

Explanation:

There are 180 degrees in every triangle. Since this triangle is a right triangle, one of the angles measures 90 degrees.

 

Therefore, 90 + 2x + \frac{x}{2}= 180.

90=2.5x

x=36

Example Question #531 : Psat Mathematics

Thingy_5

Refer to the above diagram. Which of the following gives a valid alternative name for  ?

Possible Answers:

All four of the other choices gives a valid alternative name for the triangle.

Correct answer:

All four of the other choices gives a valid alternative name for the triangle.

Explanation:

A triangle can be named after its three vertices in any order, so all of the choices given are valid.

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 acute.

The triangle cannot exist.

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 #102 : 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 : 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 #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 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 #2 : 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:

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 #1 : 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:

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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