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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find The Area Of A Parallelogram

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

$\displaystyle 32$

$\displaystyle 24$

$\displaystyle 16$

$\displaystyle 40$

Correct answer:

$\displaystyle 32$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=4 \times 12$ $\displaystyle \text{Area of Rectangle}=4 \times 12$

$\text{Area of Rectangle}=48$ $\displaystyle \text{Area of Rectangle}=48$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{4}{2}$ $\displaystyle \text{height}=\frac{4}{2}$

$\text{height}=2$ $\displaystyle \text{height}=2$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=8 \times 2$ $\displaystyle \text{Area of Parallelogram}=8 \times 2$

$\text{Area of Parallelogram}=16$ $\displaystyle \text{Area of Parallelogram}=16$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=48-16$ $\displaystyle \text{Area of Shaded Region}=48-16$

Solve.

$\text{Area of Shaded Region}=32$ $\displaystyle \text{Area of Shaded Region}=32$

Report an Error

Example Question #10 : How To Find The Area Of A Parallelogram

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

180 $\displaystyle 180$

120 $\displaystyle 120$

$\displaystyle 90$

150 $\displaystyle 150$

Correct answer:

150 $\displaystyle 150$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=12 \times 20$ $\displaystyle \text{Area of Rectangle}=12 \times 20$

$\text{Area of Rectangle}=240$ $\displaystyle \text{Area of Rectangle}=240$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{12}{2}$ $\displaystyle \text{height}=\frac{12}{2}$

$\text{height}=6$ $\displaystyle \text{height}=6$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=15 \times 6$ $\displaystyle \text{Area of Parallelogram}=15 \times 6$

$\text{Area of Parallelogram}=\90$ $\displaystyle \text{Area of Parallelogram}=\90$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=240-90$ $\displaystyle \text{Area of Shaded Region}=240-90$

Solve.

$\text{Area of Shaded Region}=150$ $\displaystyle \text{Area of Shaded Region}=150$

Report an Error

Example Question #221 : Quadrilaterals

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

182 $\displaystyle 182$

224 $\displaystyle 224$

256 $\displaystyle 256$

124 $\displaystyle 124$

Correct answer:

256 $\displaystyle 256$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=16\times 22$ $\displaystyle \text{Area of Rectangle}=16\times 22$

$\text{Area of Rectangle}=352$ $\displaystyle \text{Area of Rectangle}=352$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{16}{2}$ $\displaystyle \text{height}=\frac{16}{2}$

$\text{height}=8$ $\displaystyle \text{height}=8$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=12 \times 8$ $\displaystyle \text{Area of Parallelogram}=12 \times 8$

$\text{Area of Parallelogram}=96$ $\displaystyle \text{Area of Parallelogram}=96$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=352-96$ $\displaystyle \text{Area of Shaded Region}=352-96$

Solve.

$\text{Area of Shaded Region}=256$ $\displaystyle \text{Area of Shaded Region}=256$

Report an Error

Example Question #222 : Quadrilaterals

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

$\displaystyle 26$

$\displaystyle 24$

$\displaystyle 14$

$\displaystyle 18$

Correct answer:

$\displaystyle 18$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=4 \times 8$ $\displaystyle \text{Area of Rectangle}=4 \times 8$

$\text{Area of Rectangle}=32$ $\displaystyle \text{Area of Rectangle}=32$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{4}{2}$ $\displaystyle \text{height}=\frac{4}{2}$

$\text{height}=2$ $\displaystyle \text{height}=2$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=7\times 2$ $\displaystyle \text{Area of Parallelogram}=7\times 2$

$\text{Area of Parallelogram}=14$ $\displaystyle \text{Area of Parallelogram}=14$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=32-14$ $\displaystyle \text{Area of Shaded Region}=32-14$

Solve.

$\text{Area of Shaded Region}=18$ $\displaystyle \text{Area of Shaded Region}=18$

Report an Error

Example Question #223 : Quadrilaterals

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

28.5 $\displaystyle 28.5$

38.5 $\displaystyle 38.5$

26.5 $\displaystyle 26.5$

32.5 $\displaystyle 32.5$

Correct answer:

28.5 $\displaystyle 28.5$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=3\times12$ $\displaystyle \text{Area of Rectangle}=3\times12$

$\text{Area of Rectangle}=36$ $\displaystyle \text{Area of Rectangle}=36$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{3}{2}$ $\displaystyle \text{height}=\frac{3}{2}$

$\text{height}=1.5$ $\displaystyle \text{height}=1.5$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=5\times 1.5$ $\displaystyle \text{Area of Parallelogram}=5\times 1.5$

$\text{Area of Parallelogram}=7.5$ $\displaystyle \text{Area of Parallelogram}=7.5$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=36-7.5$ $\displaystyle \text{Area of Shaded Region}=36-7.5$

Solve.

$\text{Area of Shaded Region}=28.5$ $\displaystyle \text{Area of Shaded Region}=28.5$

Report an Error

Example Question #224 : Quadrilaterals

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

570 $\displaystyle 570$

650 $\displaystyle 650$

530 $\displaystyle 530$

610 $\displaystyle 610$

Correct answer:

570 $\displaystyle 570$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=30\times 32$ $\displaystyle \text{Area of Rectangle}=30\times 32$

$\text{Area of Rectangle}=960$ $\displaystyle \text{Area of Rectangle}=960$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{30}{2}$ $\displaystyle \text{height}=\frac{30}{2}$

$\text{height}=15$ $\displaystyle \text{height}=15$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=26\times 15$ $\displaystyle \text{Area of Parallelogram}=26\times 15$

$\text{Area of Parallelogram}=390$ $\displaystyle \text{Area of Parallelogram}=390$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=960-390$ $\displaystyle \text{Area of Shaded Region}=960-390$

Solve.

$\text{Area of Shaded Region}=570$ $\displaystyle \text{Area of Shaded Region}=570$

Report an Error

Example Question #225 : Quadrilaterals

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

280 $\displaystyle 280$

320 $\displaystyle 320$

260 $\displaystyle 260$

360 $\displaystyle 360$

Correct answer:

320 $\displaystyle 320$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=20\times 25$ $\displaystyle \text{Area of Rectangle}=20\times 25$

$\text{Area of Rectangle}=500$ $\displaystyle \text{Area of Rectangle}=500$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{20}{2}$ $\displaystyle \text{height}=\frac{20}{2}$

$\text{height}=10$ $\displaystyle \text{height}=10$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=18\times 10$ $\displaystyle \text{Area of Parallelogram}=18\times 10$

$\text{Area of Parallelogram}=180$ $\displaystyle \text{Area of Parallelogram}=180$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=500-180$ $\displaystyle \text{Area of Shaded Region}=500-180$

Solve.

$\text{Area of Shaded Region}=320$ $\displaystyle \text{Area of Shaded Region}=320$

Report an Error

Example Question #226 : Quadrilaterals

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

730 $\displaystyle 730$

850 $\displaystyle 850$

770 $\displaystyle 770$

810 $\displaystyle 810$

Correct answer:

810 $\displaystyle 810$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=40\times 30$ $\displaystyle \text{Area of Rectangle}=40\times 30$

$\text{Area of Rectangle}=1200$ $\displaystyle \text{Area of Rectangle}=1200$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{30}{2}$ $\displaystyle \text{height}=\frac{30}{2}$

$\text{height}=15$ $\displaystyle \text{height}=15$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=26\times 15$ $\displaystyle \text{Area of Parallelogram}=26\times 15$

$\text{Area of Parallelogram}=390$ $\displaystyle \text{Area of Parallelogram}=390$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=1200-390$ $\displaystyle \text{Area of Shaded Region}=1200-390$

Solve.

$\text{Area of Shaded Region}=810$ $\displaystyle \text{Area of Shaded Region}=810$

Report an Error

Example Question #227 : Quadrilaterals

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

185 $\displaystyle 185$

195 $\displaystyle 195$

205 $\displaystyle 205$

215 $\displaystyle 215$

Correct answer:

195 $\displaystyle 195$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=6\times 40$ $\displaystyle \text{Area of Rectangle}=6\times 40$

$\text{Area of Rectangle}=240$ $\displaystyle \text{Area of Rectangle}=240$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{6}{2}$ $\displaystyle \text{height}=\frac{6}{2}$

$\text{height}=3$ $\displaystyle \text{height}=3$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=15 \times 3$ $\displaystyle \text{Area of Parallelogram}=15 \times 3$

$\text{Area of Parallelogram}=45$ $\displaystyle \text{Area of Parallelogram}=45$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=240-45$ $\displaystyle \text{Area of Shaded Region}=240-45$

Solve.

$\text{Area of Shaded Region}=195$ $\displaystyle \text{Area of Shaded Region}=195$

Report an Error

Example Question #11 : How To Find The Area Of A Parallelogram

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

490 $\displaystyle 490$

530 $\displaystyle 530$

370 $\displaystyle 370$

390 $\displaystyle 390$

Correct answer:

490 $\displaystyle 490$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=31\times 28$ $\displaystyle \text{Area of Rectangle}=31\times 28$

$\text{Area of Rectangle}=868$ $\displaystyle \text{Area of Rectangle}=868$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{28}{2}$ $\displaystyle \text{height}=\frac{28}{2}$

$\text{height}=14$ $\displaystyle \text{height}=14$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=27\times 14$ $\displaystyle \text{Area of Parallelogram}=27\times 14$

$\text{Area of Parallelogram}=378$ $\displaystyle \text{Area of Parallelogram}=378$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=868-378$ $\displaystyle \text{Area of Shaded Region}=868-378$

Solve.

$\text{Area of Shaded Region}=490$ $\displaystyle \text{Area of Shaded Region}=490$

Report an Error

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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : How To Find The Area Of A Parallelogram

Example Question #10 : How To Find The Area Of A Parallelogram

Example Question #221 : Quadrilaterals

Example Question #222 : Quadrilaterals

Example Question #223 : Quadrilaterals

Example Question #224 : Quadrilaterals

Example Question #225 : Quadrilaterals

Example Question #226 : Quadrilaterals

Example Question #227 : Quadrilaterals

Example Question #11 : How To Find The Area Of A Parallelogram

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