Plane Geometry

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ISEE Upper Level Quantitative Reasoning › Plane Geometry

Questions 1 - 10

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?

Explanation

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?

Let's begin with our formula for the area of a square:

where s is our side length and A is our area.

With this formula, we can solve for our side length by plugging in our area and square rooting both sides.

$s=\sqrt{225ft^2}=15ft$

Find the circumference of a circle with a radius of 4cm.

$8 \pi \text{ cm}$

$4\pi \text{ cm}$

$12\pi \text{cm}$

$2\pi \text{ cm}$

$16\pi \text{ cm}$

Explanation

To find the circumference of a circle, we will use the following formula:

$C = 2 \cdot \pi \cdot r$

where r is the radius of the circle.

Now, we know the radius of the circle is 4cm.

Knowing this, we can substitute into the formula. We get

$C = 2 \cdot \pi \cdot 4\text{cm}$

$C = 2\pi \cdot 4\text{cm}$

$C = 8\pi \text{ cm}$

Sector SOW has a central angle of $45^{\circ}$ . What percentage of the circle does it cover?

Explanation

Sector SOW has a central angle of $45^{\circ}$ . What percentage of the circle does it cover?

Recall that there is a total of 360 degrees in a circle. SOW occupies 45 of them. To find the percentage, simply do the following:

$\frac{45^{\circ}}{360^{\circ}}*100=.125*100=12.5%$

A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?

decreased by 4%

increased by 20%

the area remains the same

decreased by 10%

Explanation

The area decreases by 20% of 20%, which is 4%.

The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.

10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.

The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?

Explanation

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?

Let's begin with our formula for the area of a square:

where s is our side length and A is our area.

With this formula, we can solve for our side length by plugging in our area and square rooting both sides.

$s=\sqrt{225ft^2}=15ft$

A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?

decreased by 4%

increased by 20%

the area remains the same

decreased by 10%

Explanation

The area decreases by 20% of 20%, which is 4%.

10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.

The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.

Find the perimeter of a hexagon with one side having a length of 15cm.

$90\text{cm}$

$120\text{cm}$

$105\text{cm}$

$75\text{cm}$

$135\text{cm}$

Explanation

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

$P = 6 \cdot a$

where a is the length of one side of the hexagon.

Now, we know the length of one side of the hexagon is 15cm.

Knowing this, we can substitute into the formula. We get

$P = 6 \cdot 15\text{cm}$

$P = 90\text{cm}$

Icecreamcone

Refer to the above figure. The shaded region is a semicircle with area $16 \pi$ . Give the perimeter of $\bigtriangleup ABC$ .

$24 \sqrt{2}$

$24 \sqrt{3}$

$12 \sqrt{2}$

$12 \sqrt{3}$

Explanation

Given the radius of a semicircle, its area can be calculated using the formula

$A = \frac{1}{2} \pi r ^{2}$ .

Substituting $A = 16 \pi$ :

$\frac{1}{2} \pi r ^{2} = 16 \pi$

$\frac{2}{\pi} \cdot \frac{1}{2} \pi r ^{2} = \frac{2}{\pi} \cdot 16 \pi$

$r ^{2} = 32$

$r = \sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4\sqrt{2} = 8 \sqrt{2}$ ; this is also the length of $\overline{BC}$ .

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle with sidelength $s = 8 \sqrt{2}$ . Its perimeter is three times this, or $3 \cdot 8 \sqrt{2} = 24 \sqrt{2}$