### All PSAT Math Resources

## Example Questions

### Example Question #1 : How To Find The Perimeter Of A Rhombus

If the area of a rhombus is 24 and one diagonal length is 6, find the perimeter of the rhombus.

**Possible Answers:**

20

24

12

8

16

**Correct answer:**

20

The area of a rhombus is found by

*A* = 1/2(*d*_{1})(*d*_{2})

where *d*_{1} and *d*_{2} are the lengths of the diagonals. Substituting for the given values yields

24 = 1/2(*d*_{1})(6)

24 = 3(*d*_{1})

8 = *d*_{1}

Now, use the facts that diagonals are perpendicular in a rhombus, diagonals bisect each other in a rhombus, and the Pythagorean Theorem to determine that the two diagonals form 4 right triangles with leg lengths of 3 and 4. Since 3^{2} + 4^{2} = 5^{2}, each side length is 5, so the perimeter is 5(4) = 20.

### Example Question #1 : How To Find The Perimeter Of A Rhombus

Note: Figure NOT drawn to scale.

Calculate the perimeter of Quadrilateral in the above diagram if:

**Possible Answers:**

Insufficient information is given to answer the question.

**Correct answer:**

, so Quadrilateral is a rhombus. Its diagonals are therefore perpendicular to each other, and the four triangles they form are right triangles. Therefore, the Pythagorean theorem can be used to determine the common sidelength of Quadrilateral .

We focus on . The diagonals are also each other's bisector, so

By the Pythagorean Theorem,

26 is the common length of the four sides of Quadrilateral , so its perimeter is .

### Example Question #1 : Quadrilaterals

A rhombus has a side length of 5. Which of the following is NOT a possible value for its area?

**Possible Answers:**

24

15

25

10

30

**Correct answer:**

30

The area of a rhombus will vary as the angles made by its sides change. The "flatter" the rhombus is (with two very small angles and two very large angles, say 2, 178, 2, and 178 degrees), the smaller the area is. There is, of course, a lower bound of zero for the area, but the area can get arbitrarily small. This implies that the correct answer would be the largest choice. In fact, the largest area of a rhombus occurs when all four angles are equal, i.e. when the rhombus is a square. The area of a square of side length 5 is 25, so any value bigger than 25 is impossible to acheive.

### Example Question #1 : How To Find The Length Of The Diagonal Of A Rhombus

In Rhombus , . If is constructed, which of the following is true about ?

**Possible Answers:**

is obtuse and isosceles, but not equilateral

is acute and scalene

obtuse and scalene

is acute and equilateral

is acute and isosceles, but not equilateral

**Correct answer:**

is acute and equilateral

The figure referenced is below.

Consecutive angles of a rhombus are supplementary - as they are with all parallelograms - so

A diagonal of a rhombus bisects its angles, so

A similar argument proves that .

Since all three angles of measure , the triangle is acute. It is also equiangular, and, subsequently, equilateral.

### Example Question #1 : How To Find An Angle In A Rhombus

In Rhombus , . If is constructed, which of the following is true about ?

**Possible Answers:**

is acute and equilateral

is right and isosceles, but not equilateral

is acute and isosceles, but not equilateral

is right and scalene

is acute and scalene

**Correct answer:**

is acute and isosceles, but not equilateral

The figure referenced is below.

The sides of a rhombus are congruent by definition, so , making isosceles. It is not equilateral, since , and an equilateral triangle must have three angles.

Also, consecutive angles of a rhombus are supplementary - as they are with all parallelograms - so

A diagonal of a rhombus bisects its angles, so

Similarly,

This makes acute.

The correct response is that is acute and isosceles, but not equilateral.

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