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ISEE Middle Level Math : Plane Geometry

Study concepts, example questions & explanations for ISEE Middle Level Math

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Example Questions

Example Question #12 : How To Find The Area Of A Triangle

Rectangles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of the green region to that of the white region.

Possible Answers:

23:7

11:4

2:1

21:9

The correct answer is not given among the other choices.

Correct answer:

11:4

Explanation:

The area of the entire rectangle is the product of its length and width, or

$120 \times 50 = 6,000$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 40 \times 80 = 1,600$

The area of the green region is therefore the difference of the two, or

6,000 - 1,600 = 4,400 .

The ratio of the area of the green region to that of the white region is

$\frac{4,400}{1,600} = \frac{4,400 \div 400}{1,600\div 400} = \frac{11}{4}$

That is, 11 to 4.

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Example Question #3 : Area Of A Triangle

A triangle has a height of 9 inches and a base that is one third as long as the height. What is the area of the triangle, in square inches?

Possible Answers:

None of these

$\frac{27}{2}$

Correct answer:

$\frac{27}{2}$

Explanation:

The area of a triangle is found by multiplying the base times the height, divided by 2.

$\text{Area} =\frac{\text{base}\times \text{height}}{2}$

Given that the height is 9 inches, and the base is one third of the height, the base will be 3 inches.

$b=h\div3=9\div3=3$

We now have both the base (3) and height (9) of the triangle. We can use the equation to solve for the area.

$\text{Area} =\frac{9\times 3}{2}$

$\text{Area} =\frac{27}{2}$

The fraction cannot be simplified.

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Example Question #5 : How To Find The Area Of A Triangle

If the angles of a triangle are equal to , , and , what is the value of ?

Possible Answers:

Correct answer:

Explanation:

Given that there are 180 degrees in a triangle,

a+2a+3a=180

6a=180

a=30

Thus, 30 is the correct answer.

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Example Question #41 : Plane Geometry

The three angles of a triangle are labeled , , and . If is $97^{\circ}$ , what is the value of X+Y ?

Possible Answers:

$83^{\circ}$

$17^{\circ}$

$73^{\circ}$

$93^{\circ}$

Correct answer:

$83^{\circ}$

Explanation:

Given that the three angles of a triangle always add up to 180 degrees, the following equation can be used:

X+Y+Z=180

X+Y+97=180

X+Y=83

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Example Question #42 : Plane Geometry

In an equilateral triangle, which of the following is NOT true?

Possible Answers:

There is a 60 degree angle

All angles are equal

There is a 90 degree angle

All sides are equal

Correct answer:

There is a 90 degree angle

Explanation:

In an equilateral triangle, all sides and angles are equal. All the angles equal 60 degrees, so there is a 60 degree angle.

Therefore, the answer choice, “There is a 90 degree angle” is not true and is the correct answer choice.

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Example Question #13 : How To Find The Area Of A Triangle

The hypotenuse of a right triangle is feet; it has one leg feet long. Give its area in square inches.

Possible Answers:

$43,120 \textrm{ in}^{2}$

$42,120 \textrm{ in}^{2}$

$77,760 \textrm{ in}^{2}$

$101,088 \textrm{ in}^{2}$

$38,880 \textrm{ in}^{2}$

Correct answer:

$38,880 \textrm{ in}^{2}$

Explanation:

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set c = 39 , b = 36 :

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

$a^{2} = 39 ^{2} - 36^{2}$

$a^{2} = 1,521 - 1,296$

$a^{2} = 225$

$a = \sqrt{225} = 15$

The legs have length and feet; multiply both dimensions by to convert to inches:

$15 \times 12 = 180$ inches

$36 \times 12 = 432$ inches.

Now find half the product:

$A = \frac{1}{2} \cdot 180 \cdot 432 = 38,880\ in^{2}$

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Example Question #14 : How To Find The Area Of A Triangle

What is the area (in square feet) of a triangle with a base of feet and a height of feet?

Possible Answers:

Correct answer:

Explanation:

The area of a triangle is found by multiplying the base times the height, divided by .

$Area =\frac{base\cdot height}{2}$

$Area =\frac{4\cdot 7}{2}$

Area=14

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Example Question #42 : Plane Geometry

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

Possible Answers:

Correct answer:

Explanation:

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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Example Question #41 : Triangles

You have two traingular gardens next to each other. They both have a base of and a height of . What is the total area?

Possible Answers:

Correct answer:

Explanation:

The area of a triangle is

$\frac{1}{2}*base*height$

and since there are two identical traingles, them put together will just be

base*height .

So your answer will just be 5*7=35 .

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Example Question #45 : Plane Geometry

Pentagon

The above figure depicts Square ABCD with perimeter 240. , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , respectively.

Give the area of Polygon XZYCD .

Possible Answers:

2,700

3,600

3,150

1,800

Correct answer:

2,700

Explanation:

Square ABCD has perimeter 240, so the length of each side is one fourth of this, or

$\frac{1}{4} \cdot 240 = 60$ .

Segment $\overline{XY}$ , as seen below, divides Polygon CDXZY into two figures:

Pentagon 1

One figure is $\bigtriangleup XZY$ , with base and height 60 and 30, respectively. Its area is half their product, or

$A_{1} = \frac{1}{2} \cdot 60 \cdot 30 = 900$

The other figure is Rect XYCD , whose length and width are 60 and 30, respectively. Its area is their product, or

$A_{2} = \cdot 60 \cdot 30 = 1,800$

Add these areas:

$A_{1} +A_{2} = 900 + 1,800 = 2,700$ .

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ISEE Middle Level Math : Plane Geometry

Study concepts, example questions & explanations for ISEE Middle Level Math

All ISEE Middle Level Math Resources

Example Questions

Example Question #12 : How To Find The Area Of A Triangle

Example Question #3 : Area Of A Triangle

Example Question #5 : How To Find The Area Of A Triangle

Example Question #41 : Plane Geometry

Example Question #42 : Plane Geometry

Example Question #13 : How To Find The Area Of A Triangle

Example Question #14 : How To Find The Area Of A Triangle

Example Question #42 : Plane Geometry

Example Question #41 : Triangles

Example Question #45 : Plane Geometry

All ISEE Middle Level Math Resources

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