# HSPT Quantitative : How to make geometric comparisons

## Example Questions

### Example Question #31 : Geometric Comparison

Which of the following is the closest to 25?

a. Perimeter of a square with side length

b. Diameter of a circle with radius

c. Perimeter of an equilateral triangle with side length

Both and

Both  and

Explanation:

Find the values of a, b, and c.

Since is closest to , the answer is c.

### Example Question #32 : Geometric Comparison

Find the correct relationship between the perimeters of these shapes.

a. A rhombus of side length 4

b. A regular hexagon of side length 2

c. A regular pentagon of side length 3

Explanation:

Find the perimeters of the shapes.

A rhombus has four identical sides:

A hexagon has six:

A pentagon has five:

The correct relationship is .

### Example Question #33 : Geometric Comparison

, and  each refer to the fractions of their respective figures that are gray.

Which statement is correct?

Explanation:

Examine the diagram below, which shows all three figures with additional lines.

is represented by four colored portions out of eight, so .

is represented by six colored portions out of twelve, so

is represented by four colored portions out of six, so .

. Since

and

,

it follows that .

### Example Question #34 : Geometric Comparison

Figures NOT drawn to scale

Let , and  refer to the diameters of their respective circles.

Which is a true statement?

Explanation:

Circle  has radius 4; its diameter is twice this, which is 8.

Circle  has area . Since the area is equal to  times the square of the radius, that is,

,

substitute  for the area and solve for :

The diameter of the circle is twice this, or 4.

Circle  has circumference ; its diameter is the circumference divided by , which is

Therefore, .

### Example Question #35 : Geometric Comparison

Note: Figures NOT drawn to scale.

In the above diagram, let , and  represent the radii of their respective circles.

Which is a correct statement?

Explanation:

The radius of a circle is one half its diameter. Circle A has diameter 16, so its radius is half this, or 8.

The relationship between the area of a circle and its radius  is given by the equation

Circle B has area , so substitute this for the area:

The radius of a circle is its circumference divided by . Circle C has circumference , so divide this by :

and , so .

### Example Question #36 : Geometric Comparison

, and  each refer to the fractions of their respective figures that are gray.

Which statement is correct?

Explanation:

Examine the diagram below, which shows all three figures with additional lines.

Note that each square has been divided into eight congruent parts. , and  are represented by 3, 4, and 5 colored portions, respectively, so

,

and .

Of the four choices, only  is correct.

### Example Question #37 : Geometric Comparison

Examine the above three figures. All three squares have the same area.

Of Figure B and Figure C, which figure(s) have the same area shaded in as does Figure A?

Neither B nor C

C but not B

B but not C

Both B and C

C but not B

Explanation:

Figure A is divided into twelve regions of equal area; six are shaded in, which make up  of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 6:

.

Figure B is divided into sixteen regions of equal area; seven are shaded in, which make up  of the area. This is in lowest terms, since 7 and 12 are relatively prime. , so Figure B does not have the same shaded area as Figure A.

Figure C is divided into eight regions of equal area; four are shaded in, which make up  of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 4:

.

Figure C has the same shaded area as Figure A.

The correct response is "Figure C, but not Figure B."

### Example Question #38 : Geometric Comparison

Examine the above three figures. All three squares have the same area.

Of Figure B and Figure C, which figure(s) have the same area shaded in as does Figure A?

Both Figure B and Figure C

Figure C but not Figure B

Neither Figure B nor Figure C

Figure B but not figure C

Both Figure B and Figure C

Explanation:

Figure A is divided into twelve regions of equal area; nine are shaded in, which make up  of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 3:

.

Figure B is divided into sixteen regions of equal area; twelve are shaded in, which make up  of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 4:

Figure B has the same shaded area as Figure A.

Figure C is divided into eight regions of equal area; six are shaded in, which make up  of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 4:

.

Figure C has the same shaded area as Figure A.

The correct response is "Both Figure B and Figure C."

### Example Question #39 : Geometric Comparison

Refer to the above diagram. Let , and  stand for the fraction of their respective squares that are shaded in. Which is correct?

Explanation:

Below are the same figures, but with some additional lines drawn so that each square is divided into equal parts.

is represented by 6 parts out of 12, so .

is represented by 4 parts out of 9, so .

is represented by 3 parts out of 8, so .

These fractions can be compared by expressing them in terms of a least common denominator - :

Comparing numerators, we see that .

### Example Question #40 : Geometric Comparison

Examine (a), (b), and (c) to find the best answer:

a) perimeter of a square with a side length of

b) perimeter of a rectangle with a length of  and a width of

c) perimeter of an equailateral triangle with a side length of