# PSAT Math : Geometry

## Example Questions

### Example Question #1 : Quadrilaterals

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

24

15

25

10

30

30

Explanation:

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 ?

is obtuse and isosceles, but not equilateral

is acute and equilateral

is acute and isosceles, but not equilateral

is acute and scalene

obtuse and scalene

is acute and equilateral

Explanation:

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 ?

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

is acute and isosceles, but not equilateral

Explanation:

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.

### Example Question #1 : How To Find If Quadrilaterals Are Similar

Quadrilateral ABCD contains four ninety-degree angles. Which of the following must be true?

I. Quadrilateral ABCD is a rectangle.

II. Quadrilateral ABCD is a rhombus.

III. Quadrilateral ABCD is a square.

I, II, and III

I only

II only

I and II only

II and III only

I only

Explanation:

Quadrilateral ABCD has four ninety-degree angles, which means that it has four right angles because every right angle measures ninety degrees. If a quadrilateral has four right angles, then it must be a rectangle by the definition of a rectangle. This means statement I is definitely true.

However, just because ABCD has four right angles doesn't mean that it is a rhombus. In order for a quadrilateral to be considered a rhombus, it must have four congruent sides. It's possible to have a rectangle whose sides are not all congruent. For example, if a rectangle has a width of 4 meters and a length of 8 meters, then not all of the sides of the rectangle would be congruent. In fact, in a rectangle, only opposite sides need be congruent. This means that ABCD is not necessarily a rhombus, and statement II does not have to be true.

A square is defined as a rhombus with four right angles. In a square, all of the sides must be congruent. In other words, a square is both a rectangle and a rhombus. However, we already established that ABCD doesn't have to be a rhombus. This means that ABCD need not be a square, because, as we said previously, not all of its sides must be congruent. Therefore, statement III isn't necessarily true either.

The only statement that has to be true is statement I.

### Example Question #2 : How To Find The Area Of A Trapezoid

A trapezoid has a base of length 4, another base of length s, and a height of length s. A square has sides of length s. What is the value of s such that the area of the trapezoid and the area of the square are equal?

Explanation:

In general, the formula for the area of a trapezoid is (1/2)(a + b)(h), where a and b are the lengths of the bases, and h is the length of the height. Thus, we can write the area for the trapezoid given in the problem as follows:

area of trapezoid = (1/2)(4 + s)(s)

Similarly, the area of a square with sides of length a is given by a2. Thus, the area of the square given in the problem is s2.

We now can set the area of the trapezoid equal to the area of the square and solve for s.

(1/2)(4 + s)(s) = s2

Multiply both sides by 2 to eliminate the 1/2.

(4 + s)(s) = 2s2

Distribute the s on the left.

4s + s2 = 2s2

Subtract s2 from both sides.

4s = s2

Because s must be a positive number, we can divide both sides by s.

4 = s

This means the value of s must be 4.

### Example Question #2 : Quadrilaterals

Note: Figure NOT drawn to scale.

The white region in the above diagram is a trapezoid. What percent of the above rectangle, rounded to the nearest whole percent, is blue?

Explanation:

The area of the entire rectangle is the product of its length and width, or

.

The area of the white trapezoid is one half the product of its height and the sum of its base lengths, or

Therefore, the blue polygon has area

This is

of the rectangle.

Rounded, this is 70%.

### Example Question #611 : Psat Mathematics

Refer to the above diagram. .

Give the area of Quadrilateral .

Explanation:

, since both are right; by the Corresponding Angles Theorem,  , and Quadrilateral  is a trapezoid.

By the Angle-Angle Similarity Postulate, since

and

(by reflexivity),

and since corresponding sides of similar triangles are in proportion,

, the larger base of the trapozoid;

The smaller base is .

, the height of the trapezoid.

The area of the trapezoid is

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

A circle with a radius 2 in is inscribed in a square. What is the perimeter of the square?

24 in

32 in

12 in

28 in

16 in

16 in

Explanation:

To inscribe means to draw inside a figure so as to touch in as many places as possible without overlapping. The circle is inside the square such that the diameter of the circle is the same as the side of the square, so the side is actually 4 in.  The perimeter of the square = 4s = 4 * 4 = 16 in.

### Example Question #2 : How To Find The Perimeter Of A Square

Square X has 3 times the area of Square Y.  If the perimeter of Square Y is 24 ft, what is the area of Square X, in sq ft?

54

108

144

112

72

108

Explanation:

Find the area of Square Y, then calculate the area of Square X.

If the perimeter of Square Y is 24, then each side is 24/4, or 6.

A = 6 * 6 = 36 sq ft, for Square Y

If Square X has 3 times the area, then 3 * 36 = 108 sq ft.

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

A square has an area of .  If the side of the square is reduced by a factor of two, what is the perimeter of the new square?