ISEE Upper Level Quantitative Reasoning - Plane Geometry

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.

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

Trapezoid

In the above figure, $\overline{XY}$ is the midsegment of Trapezoid . Give the ratio of the area of Trapezoid to that of Trapezoid .

33 to 19

10 to 3

13 to 6

20 to 13

Explanation

Midsegment $\overline{XY}$ divides Trapezoid into two trapezoids of the same height, which we will call ; the length of the midsegment is half sum of the lengths of the bases:

$XY = \frac{1}{2} (12 + 40) = \frac{1}{2} \cdot 52 = 26$ .

The area of a trapezoid is one half multiplied by its height multiplied by the sum of the lengths of its bases. Therefore, the area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (TR+XY ) = \frac{1}{2} \cdot h \cdot (12+26 ) = \frac{1}{2} \cdot 38 \cdot h = 19 h$

The area of Trapezoid is

$\frac{1}{2} \cdot h \cdot (XY +AP) = \frac{1}{2} \cdot h \cdot ( 26+40 ) = \frac{1}{2} \cdot 66 \cdot h = 33 h$

The ratio of the areas is

$\frac{33 h}{19h} = \frac{33}{19}$ , or 33 to 19.

In Parallelogram , and . Which of the following is greater?

(A)

(B)

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

Explanation

In Parallelogram , $\overline{AB}$ and $\overline{BC }$ are adjoining sides; there is no specific rule for the relationship between their lengths. Therefore, no conclusion can be drawn of and , and no conclusion can be drawn of the relationship between and .

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 $3 \cdot 8 \sqrt{2} = 24 \sqrt{2}$

The lengths of the sides of ten squares form an arithmetic sequence. One side of the smallest square measures sixty centimeters; one side of the second-smallest square measures one meter.

Give the area of the largest square, rounded to the nearest square meter.

18 square meters

16 square meters

20 square meters

22 square meters

24 square meters

Explanation

Let $a_{n}$ be the lengths of the sides of the squares in meters. $a_{1} = 60 \div 100 = 0.6$ and $a_{2} = 1$ , so their common difference is

The arithmetic sequence formula is

$a_{n} = a_{1} + (n-1)d$

The length of a side of the largest square - square 10 - can be found by substituting $a_{1} = 0.6, n= 10, d = 0.4$ :

$a_{10} =0.6+ (10-1) 0.4 = 0.6 + 9 (0.4) = 0.6 + 3.6 = 4.2$

The largest square has sides of length 4.2 meters, so its area is the square of this, or $A = 4.2^{2} = 17.64$ square meters.

Of the choices, 18 square meters is closest.

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

$154.8^{\circ}$

$15.5^{\circ}$

$430^{\circ}$

$366^{\circ}$

Explanation

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

Use the following formula and solve for x:

$\frac{x}{360^{\circ}}100=43%$

Begin by dividing over the 100

$\frac{x}{360^{\circ}}=.43$

Then multiply by 360

$x=.43*360^{\circ}=154.8^{\circ}$

Trapezoid

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

(a) $m \angle T + m \angle P$

(b) $m \angle R + m \angle A$

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

Explanation

$\overline{TR} \parallel \overline{PA}$ ; $\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always $180^{\circ}$ .

Therefore, $m \angle T + m \angle P = 180= m \angle R + m \angle A$ , making the two quantities equal.

$\Delta ABC$ and $\Delta DEF$ are right triangles, with right angles $\angle B , \angle E$ , respectively.

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of $\Delta DEF$

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Explanation

No information is given about the legs of either triangle; therefore, no information about their perimeters can be deduced.

The circumferences of eight circles form an arithmetic sequence. The smallest circle has radius two inches; the second smallest circle has radius five inches. Give the radius of the largest circle.

1 foot, 11 inches

2 feet

2 feet, 1 inch

4 feet 2 inches

3 feet 10 inches

Explanation

The circumference of a circle can be determined by multiplying its radius by $2 \pi$ , so the circumferences of the two smallest circles are

$C_{1} = 2 \cdot 2 \pi = 4 \pi$

and

$C_{2} = 2 \cdot 5 \pi = 10 \pi$

The circumferences form an arithmetic sequence with common difference

$C_{2}- C_{1}= 10 \pi - 4 \pi = 6 \pi$

The circumference of a circle can therefore be found using the formula

$C_{n} = C_{1} + (n-1)d$

where $C_{1} = 4 \pi$ and $d = 6 \pi$ ; we are looking for that of the th smallest circle, so

$C_{8} = 4 \pi + (8-1)6 \pi$

$= 4 \pi + 7 \cdot 6 \pi$

$= 4 \pi + 42 \pi$

$= 46 \pi$

Since the radius of a circle is the circumference of the circle divided by $2 \pi$ , the radius of this eighth circle is

$r = \frac{46 \pi}{2 \pi} = 23$ inches, or 1 foot 11 inches.

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.

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

Plane Geometry

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