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

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.

3

Possible Answers:

\(\displaystyle 32\)

\(\displaystyle 24\)

\(\displaystyle 16\)

\(\displaystyle 40\)

Correct answer:

\(\displaystyle 32\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=48\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{4}{2}\)

\(\displaystyle \text{height}=2\)

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

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

\(\displaystyle \text{Area of Parallelogram}=16\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=32\)

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.

4

Possible Answers:

\(\displaystyle 180\)

\(\displaystyle 120\)

\(\displaystyle 90\)

\(\displaystyle 150\)

Correct answer:

\(\displaystyle 150\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=240\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{12}{2}\)

\(\displaystyle \text{height}=6\)

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

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

\(\displaystyle \text{Area of Parallelogram}=\90\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=150\)

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.

5

Possible Answers:

\(\displaystyle 182\)

\(\displaystyle 224\)

\(\displaystyle 256\)

\(\displaystyle 124\)

Correct answer:

\(\displaystyle 256\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=352\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{16}{2}\)

\(\displaystyle \text{height}=8\)

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

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

\(\displaystyle \text{Area of Parallelogram}=96\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=256\)

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.

6

Possible Answers:

\(\displaystyle 26\)

\(\displaystyle 24\)

\(\displaystyle 14\)

\(\displaystyle 18\)

Correct answer:

\(\displaystyle 18\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=32\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{4}{2}\)

\(\displaystyle \text{height}=2\)

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

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

\(\displaystyle \text{Area of Parallelogram}=14\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=18\)

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.

7

Possible Answers:

\(\displaystyle 28.5\)

\(\displaystyle 38.5\)

\(\displaystyle 26.5\)

\(\displaystyle 32.5\)

Correct answer:

\(\displaystyle 28.5\)

Explanation:

13

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:

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

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

\(\displaystyle \text{Area of Rectangle}=3\times12\)

\(\displaystyle \text{Area of Rectangle}=36\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{3}{2}\)

\(\displaystyle \text{height}=1.5\)

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

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

\(\displaystyle \text{Area of Parallelogram}=7.5\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=28.5\)

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.

9

Possible Answers:

\(\displaystyle 570\)

\(\displaystyle 650\)

\(\displaystyle 530\)

\(\displaystyle 610\)

Correct answer:

\(\displaystyle 570\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=960\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{30}{2}\)

\(\displaystyle \text{height}=15\)

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

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

\(\displaystyle \text{Area of Parallelogram}=390\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=570\)

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.

8

Possible Answers:

\(\displaystyle 280\)

\(\displaystyle 320\)

\(\displaystyle 260\)

\(\displaystyle 360\)

Correct answer:

\(\displaystyle 320\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=500\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{20}{2}\)

\(\displaystyle \text{height}=10\)

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

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

\(\displaystyle \text{Area of Parallelogram}=180\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=320\)

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.

10

Possible Answers:

\(\displaystyle 730\)

\(\displaystyle 850\)

\(\displaystyle 770\)

\(\displaystyle 810\)

Correct answer:

\(\displaystyle 810\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=1200\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{30}{2}\)

\(\displaystyle \text{height}=15\)

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

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

\(\displaystyle \text{Area of Parallelogram}=390\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=810\)

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.

11

Possible Answers:

\(\displaystyle 185\)

\(\displaystyle 195\)

\(\displaystyle 205\)

\(\displaystyle 215\)

Correct answer:

\(\displaystyle 195\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=240\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{6}{2}\)

\(\displaystyle \text{height}=3\)

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

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

\(\displaystyle \text{Area of Parallelogram}=45\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=195\)

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.

12

Possible Answers:

\(\displaystyle 490\)

\(\displaystyle 530\)

\(\displaystyle 370\)

\(\displaystyle 390\)

Correct answer:

\(\displaystyle 490\)

Explanation:

13

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:

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

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

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

\(\displaystyle \text{Area of Rectangle}=868\)

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

\(\displaystyle \text{height}=\frac{\text{length}}{2}\)

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

\(\displaystyle \text{height}=\frac{28}{2}\)

\(\displaystyle \text{height}=14\)

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

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

\(\displaystyle \text{Area of Parallelogram}=378\)

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.

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

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

Solve.

\(\displaystyle \text{Area of Shaded Region}=490\)

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