Plane Geometry - ISEE Middle Level Quantitative Reasoning

Card 0 of 990

Question

A regular pentagon has perimeter 1 yard. Give the length of one side.

Answer

A regular pentagon has five sides of equal length. The perimeter, which is the sum of the lengths of these sides, is one yard, which is equal to 36 inches. Therefore, the length of one side is

$36 \textrm{ in} \div 5 = 7\frac{1}{5} \textrm{ in}$ .

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Question

Note: Figure NOT drawn to scale

The above figure shows Square .

Which is the greater quantity?

(a) The area of Trapezoid

(b) The area of Trapezoid

Answer

The easiest way to answer the question is to locate on $\overline{SQ}$ such that :

Trapezoids and have the same height, which is . Their bases, by construction, have the same lengths - and . Therefore, Trapezoids and have the same area.

Since , it follows that , and . It follows that Trapezoid is greater in area than Trapezoids and , and Trapezoid is less in area.

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Question

A regular hexagon has perimeter 8 feet. Give the length of one side.

Answer

The perimeter can be converted from feet to inches by multiplying by conversion factor 12 {inches per foot):

$8 \textrm{ ft} \times 12 \textrm{ in/ft} = 96 \textrm{ in}$

A regular hexagon has six sides of equal length. Divide this perimeter by 6 to obtain the length of each side:

$96 \textrm{ in} \div 6 = 16 \textrm{ in}$

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Question

A triangle has base 80 inches and area 4,200 square inches. What is its height?

Answer

Use the area formula for a triangle, setting :

$A = \frac{1}{2} bh$

$4200 = \frac{1}{2}\cdot 80 \cdot h = 40h$

$h = 4200 \div 40 = 105$ inches

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Question

The sum of the lengths of the legs of an isosceles right triangle is one meter. What is its area in square centimeters?

Answer

The legs of an isosceles right triangle have equal length, so, if the sum of their lengths is one meter, which is equal to 100 centimeters, each leg measures half of this, or

$100 \div 2 = 50$ centimeters.

The area of a triangle is half the product of its height and base; for a right triangle, the legs serve as height and base, so the area of the triangle is

$\frac{1}{2} \times 50 \times 50 = 1,250$ square centimeters.

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Question

Figure NOT drawn to scale

Square has area 1,600. ; . Which of the following is the greater quantity?

(a) The area of $\bigtriangleup AZX$

(b) The area of $\bigtriangleup BZY$

Answer

Square has area 1,600, so the length of each side is $\sqrt{1,600 }= 40$ .

Since ,

Therefore, .

$\bigtriangleup AZX$ has as its area $\frac{1}{2} \cdot AX \cdot AZ$ ; $\bigtriangleup BZY$ has as its area $\frac{1}{2} \cdot BX \cdot BZ$ .

Since and , it follows that

$BX \cdot BZ > AX \cdot AZ$

and

$\frac{1}{2} \cdot BX \cdot BZ >\frac{1}{2} \cdot AX \cdot AZ$

$\bigtriangleup BZY$ has greater area than $\bigtriangleup AZX$ .

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Question

The above figure depicts Square . , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , respectively.

$\bigtriangleup AZX$ has area . What is the area of Square ?

Answer

Since , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , if we call the length of each side of the square, then

$AX = AZ = \frac{1}{2}s$

The area of $\bigtriangleup AZX$ is half the product of the lengths of its legs:

$\frac{1}{2} \cdot AX \cdot AZ$

$= \frac{1}{2} \cdot \frac{1}{2}s \cdot \frac{1}{2}s$

$= \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\cdot s \cdot s$

$= \frac{1 \cdot 1 \cdot 1}{2\cdot 2 \cdot 2}\cdot s ^{2}$

$= \frac{1}{8} s ^{2}$

The area of the square is the square of the length of a side, which is $s^{2}$ . This is eight times the area of $\bigtriangleup AZX$ , so the correct choice is

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Question

Which of the following is the greater quantity?

(a) The area of the above triangle

(b) 800

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So

$A = \frac{1}{2} \cdot 25 \cdot 60 = 750$

which is less than 800.

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Question

The above figure gives the lengths of the three sides of the triangle in feet. Give its area in square inches.

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are feet and feet.

Multiply each length by 12 to convert to inches - the lengths become and . The area in square inches is therefore

$\frac{1}{2} (12A )(12 B) = \frac{1}{2} \cdot 12 \cdot 12 \cdot A \cdot B = 72AB$ square inches.

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Question

Refer to the above figure. Which is the greater quantity?

(a) The perimeter of the triangle

(b) 3 feet

Answer

The perimeter of the triangle - the sum of the lengths of its sides - is

inches.

3 feet are equivalent to $3 \times 12 = 36$ inches, so this is the greater quantity.

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Question

Figure NOT drawn to scale.

In the above diagram, Square has area 400. Which is the greater quantity?

(a) The area of $\bigtriangleup SQE$

(b) The area of $\bigtriangleup UAR$

Answer

Square has area 400, so its common sidelength is the square root of 400, or 20. Therefore,

.

The area of a right triangle is half the product of the lengths of its legs.

$\bigtriangleup SQE$ has legs $\overline{SQ}$ and $\overline{SE}$ , so its area is

$\frac{1}{2} \cdot SQ \cdot SE = \frac{1}{2} \cdot 9 \cdot 20 = 90$ .

$\bigtriangleup UAR$ has legs $\overline{UA}$ and $\overline{AR}$ , so its area is

$\frac{1}{2} \cdot AR \cdot UA = \frac{1}{2} \cdot 11 \cdot 20 = 110$ .

$\bigtriangleup UAR$ has the greater area.

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Question

Figure NOT drawn to scale

The above diagram depicts Parallelogram . Which is the greater quantity?

(a) The area of $\bigtriangleup AMD$

(b) The area of $\bigtriangleup BMC$

Answer

Opposite sides of a parallelogram have the same measure, so

Base $\overline{AM}$ of $\bigtriangleup AMD$ and base $\overline{MB}$ of $\bigtriangleup BMC$ have the same length; also, as can be seen below, both have the same height, which is the height of the parallelogram.

Therefore, the areas of $\bigtriangleup AMD$ and $\bigtriangleup BMC$ have the same area - $\frac{1}{2} \cdot 60 \cdot h = 30h$ .

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Question

Which is the greater quantity?

(a) 370 meters

(b) 3,700 centimeters

Answer

One meter is equivalent to 100 centimeters, so 370 meters can be converted to centimeters by multiplying by 100:

$370 \times 100 = 37,000$ .

370 meters are equal to 37.000 centimeters and is the greater quantity.

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Question

Which is the greater quantity?

(a) 4.8 kilometers

(b) 4,800 meters

Answer

One kilometer is equal to 1,000 meters, so convert 4.8 kilometers to meters by multiplying by 1,000:

$4.8 \times 1,000 = 4,800$ meters

The two quantities are equal.

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Question

Which is the greater quantity?

(a) The sidelength of a cube with surface area $600 \textrm{ cm}^{2}$

(b) The sidelength of a cube with volume $1,000 \textrm{ cm}^{3}$

Answer

(a) A cube has six faces, each a square. Since the surface area of this cube is $600 \textrm{ cm}^{2}$ , each face has one-sixth this area, or $100 \textrm{ cm}^{2}$ ; the sidelength is the square root of this, or $10 \textrm{ cm}$ .

(b) The volume of a cube is the cube of its sidelength, so we take the cube root of the volume of this cube to get the sidelength:

$\sqrt[3]{1,000} = 10 \textrm{ cm}$

The cubes have the same sidelength.

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Question

Which is the greater quantity?

(a) The volume of a rectangular prism with length 60 centimeters, width 30 centimeters, and height 15 centimeters

(b) The volume of a cube with sidelength 300 millimeters

Answer

(a) The volume of the prism is the product of its length, its width, and its height:

$V = 60 \times 30 \times 15 = 27,000 \textrm{ cm}^{3}$

(b) The volume of the cube is the cube of its sidelength. We restate 300 millimeters as 30 centimeters, and cube this:

$V = 30 ^{3} = 30 \times 30 \times 30 = 27,000 \textrm{ cm}^{3}$

The volumes are equal.

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Question

Give the area of the above triangle.

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So

$A = \frac{1}{2} \cdot 25 \cdot 60 = 750$

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Question

If a parallelogram has side lengths of and , what is the area?

Answer

To find the area of a parallelogram, you use the formula,

.

Since the height in this problem is not known, you cannot solve for area.

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Question

What is the area of the triangle?

Answer

Area of a triangle can be determined using the equation:

$\small A=\frac{1}{2}bh$

$\small A=\frac{1}{2}(14)(5)=35$

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Question

What is the area of a triangle with a base of and a height of ?

Answer

The formula for the area of a triangle is $\dpi{100} Area=\frac{1}{2}\times base\times height$ .

Plug the given values into the formula to solve:

$\dpi{100} Area=\frac{1}{2}\times 12\times 3$

$\dpi{100} Area=\frac{1}{2}\times 36$

$\dpi{100} Area=18$

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