Plane Geometry - ISEE Middle Level Quantitative Reasoning

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Question

A ladder is leaning on a wall. It is ft long. The bottom of the ladder is from the base. How far up the wall is the top of the ladder?

Answer

The Pythagorean Theorem states that:

$a^{2} +b^{2} = c^{2}$

With and representing the measurement of the legs and representing the hypotenuse.

$10^{2} +b^{2} = 16^{2}$

$100 + b^{2} = 256$

$(100-100) + b^{2} = 256-100$

$b^{2} = 156$

$\sqrt{b^{2}} = \sqrt{156}$

$b\approx 12.5ft$

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Question

What is the perimeter of a right triangle if the hypotenuse is and the measurement of one of its legs is

Answer

First, use the Pythagorean Theorem to get the measurement of the other leg.

$a^{2} +b^{2} =c^{2}$

$9^{2} + b^{2} =15^{2}$

$81 + b^{2} = 225$

$(81-81) + b^{2} = 225-81$

$b^{2} = 144$

To get the perimeter of the triangle, add the measurements of each of the three sides.

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Question

Find the length of a line from the point to the point .

Answer

Find the length of a line from the point to the point .

To find this distance, we need to use distance formula (which is really similar to Pythagorean Theorem)

Distance formula is as follows

$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

Where our x's and y's come from our ordered pairs.

So, let's plug and chug

$d=\sqrt{(9--12)^2+(14-36)^2}$

Simplify

$d=\sqrt{(21)^2+(-22)^2}$

$d=\sqrt{441+484}=\sqrt{925}\approx30.4$

So our answer is 30.4

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Question

What is the length of a line segment with end points and ?

Answer

The length of a line segment can be determined using the distance formula:

$D=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

$D=\sqrt{(3-6)^2+(5-1)^2}$

$D=\sqrt{-3^2+4^2}$

$D=\sqrt{9+16}=\sqrt{25}=5$

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Question

Give the volume of the box in cubic inches.

Answer

Use the volume formula, substituting :

$V=22 \cdot 12 \cdot 12 = 3168^{3}$

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Question

Give the volume of the above box in cubic centimeters.

Answer

100 centimeters make one meter, so convert each of the dimensions of the box by multiplying by 100.

$0.9 \times 100 = 90$ centimeters

$0.6 \times 100 = 60$ centimeters

Use the volume formula, substituting :

$V = 90 \cdot 60 \cdot 60 = 324,000$ cubic centimeters

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Question

A cube made of nickel has sidelength 20 centimeters, Nickel has a density of 8.9 grams per cubic centimeter. What is the mass of this cube?

Answer

The volume of the cube is $20 ^{3} = 20 \times 20 \times 20 = 8,000$ cubic centimeters. Multiply by the number of grams per cubic centimeter to get $8,000 \times 8.9 = 71,200$ grams, or $71,200 \div 1,000 = 71.2$ kilograms.

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Question

A rectangular prism has length 24 inches, width 18 inches, and height 15 inches. Give its volume in cubic feet.

Answer

Divide each dimension in inches by 12 to convert from inches to feet:

$L = 24 \div 12 = 2$ feet

$W = 18 \div 12 = 1 \frac{1}{2} = \frac{3}{2}$ feet

$H = 15 \div 12 = 1 \frac{1}{4} = \frac{5}{4}$ feet

Multiply the three to get the volume:

$V = LWH = 2 \cdot \frac{3}{2} \cdot \frac{5}{4} = \frac{15}{4} = 3 \frac{3}{4}$ cubic feet

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Question

The above depicts a rectangular swimming pool for an apartment. The pool is six feet deep everywhere. What is the volume of the pool in cubic feet?

Answer

The pool can be seen as a rectangular prism with dimensions 50 feet by 35 feet by 6 feet; its volume is

$50 \times 35 \times 6 = 10,500$ cubic feet.

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Question

One cubic foot is equal to (about) 7.5 gallons.

The above depicts a rectangular swimming pool for an apartment. The pool is seven feet deep everywhere. How many gallons of water does the pool hold?

Answer

The pool can be seen as a rectangular prism with dimensions 50 feet by 35 feet by 7 feet; its volume in cubic feet the product of these dimensions, which is

$50 \times 35 \times 7 =12,250$ cubic feet.

One cubic foot is equal to about 7.5 gallons, so multiply:

$12,250 \times 7.5 = 91,875$ gallons.

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Question

The perimeter of a square is 56in. Find the length of one side.

Answer

To find the perimeter of a square, we use this formula

where a is the length of one side. A square has 4 equal sides. That is why we multiply the length of one side by 4. To find the length of one side, we will solve for a.

Now, we know the perimeter of the square is 56in. So, we can substitute. We get

$56\text{in} =4a$

$\frac{56\text{in}}{4} = \frac{4a}{4}$

$14\text{in} = a$

$a=14\text{in}$

Therefore, the length of one side of the square is 14in.

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Question

Sandy wants to put a border around her son’s nursery. If all four square walls in the room have the same width and use up $\small 36$ feet of border, what is the length of one wall?

Answer

When Sandy puts the border around her son's room, she will need enough to cover the perimeter. Since the room has four walls equal in length, we know that the room is a square. The perimeter of a square can by found by adding all the sides together, or by multiplying the length of one side by 4. This can be written as:

$\small \small 4s=P$

Since we know that Sandy used $\small 36$ feet of border, we know the perimeter is $\small 36$ . We can now write an equation:

$\small 4s=36$

Now, in order to isolate the variable, we can divide both sides by four.

The left-hand side simplifies to:

$\small \small \frac{4s}{4}=s$

The right-hand side simplifies to:

$\small \frac{36}{4}=9$

$\small s=9 : feet$

When we solve, we find that the length of each wall is $\small 9 : feet$ .

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Question

Use this image for the following problem.

What is the perimeter of the square in this picture?

Answer

The question only is looking for a part of the picture, just the square. With squares, the rule is that all the sides are equivalent, meaning the same lengths and all angles are right angles.

Perimeter means adding up all the sides together. So we just need to add the lengths of the sides of the square. Uh oh, we only have one side that is listed.

Again, remember that with squares the sides are equivalent, and we know one side is 5 inches. We just need to take because a square has 4 sides.

Our perimeter is .

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Question

Find the perimenter:

Answer

The perimeter is equal to the sum of the length of all sides. Each side is equal to $\small 9$ . Therefore, the perimeter equals:

$\small P=9+9+9+9=36$

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Question

Find the perimeter of a square with side length 10.

Answer

To solve, simply multiply the side length by 4 since all sides in a square area equal. Thus,

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Question

A new building in New York City will have a footprint that is a perfect square, taking up a complete city block. If one city block is of a mile, what is the perimeter of the building?

Answer

A new building in New York City will have a footprint that is a perfect square, taking up a complete city block. If one city block is of a mile, what is the perimeter of the building?

To find the perimeter of a square, we need to multiply the side length by 4.

Our side length is mile, so do the following to find our answer:

$4*\frac{1}{8}=\frac{4}{8}=\frac{1}{2}mi$

So the perimeter is $\frac{1}{2}mi$

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Question

Use the following to answer the question.

Find the perimeter of the square.

Answer

To find the perimeter of a square, we use the following formula:

$\text{perimeter of square} = a+b+c+d$

where a, b, c, and d are the lengths of each side.

Now, we know that a square has equal sides. So given the square

we know that each side equals 6ft. So, we can use this to substitute into the formula. We get

$\text{perimeter of square} = 6\text{ft} + 6\text{ft} + 6\text{ft} + 6\text{ft}$

$\text{perimeter of square} = 24\text{ft}$

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Question

Find the perimeter of a square with a width of 8 inches.

Answer

To find the perimeter of a square, we use the following formula:

$\text{perimeter of square} = a+b+c+d$

where a, b, c, and d are the lengths of the sides of the square.

Now, we know the width of the square is 8 inches. Because it is a square, we know all sides are equal. Therefore, all sides are 8 inches. Knowing this, we can substitute into the formula. We get

$\text{perimeter of square} = 8\text{ inches} +8\text{ inches} +8\text{ inches} +8\text{ inches}$

$\text{perimeter of square} = 32 \text{ inches}$

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Question

You are playing Monopoly and notice that the dice is a cube. If the dice has a height of 2 centimeters, what is the perimeter of one of its surfaces?

Answer

You are playing Monopoly and notice that the dice is a cube. If the dice has a height of 2 centimeters, what is the perimeter of one of its surfaces?

Begin by realizing that we need to find the perimeter of a square.

Next, recall that the height of a cube is the same as its length and width. In other words, one surface of a cube will be a square with sides equal to the height of the cube.

Armed with this knowledge, we can find the perimeter of the square.

So our answer is 8cm

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Question

A square stamp has an edge length of 6 centimeters, what is its perimeter?

Answer

A square stamp has an edge length of 6 centimeters, what is its perimeter?

To find the perimeter of a square, we simply can multiply the edge length by 4.

Doing so yields:

$P_{square}=6cm*4=24cm$

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