Intermediate Geometry : Triangles

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #141 : Triangles

In the figure, a triangle that shares its base with the width of the rectangle has a height that is half the length of the rectangle. Find the area of the shaded region.

7

Possible Answers:

\(\displaystyle \frac{227}{2}\)

\(\displaystyle \frac{235}{2}\)

\(\displaystyle \frac{215}{2}\)

\(\displaystyle \frac{225}{2}\)

Correct answer:

\(\displaystyle \frac{225}{2}\)

Explanation:

13

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

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 width to find the area.

\(\displaystyle \text{Area of Rectangle}=10 \times 15\)

\(\displaystyle \text{Area of Rectangle}= 150\)

Next, recall how to find the area of a triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times\text{base}\times\text{height}\)

From the question, we know that the height of the triangle is half the length of the rectangle. Use the length of the rectangle to find the height of the triangle:

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

\(\displaystyle \text{height}=\frac{10}{2}\)

\(\displaystyle \text{height}=5\)

Since the base of the triangle and the width of the rectangle are the same, we can find the area of the triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times 5\times15\)

\(\displaystyle \text{Area of Triangle}=\frac{75}{2}\)

To find the area of the shaded region, subtract the area of the triangle from the area of the rectangle.

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

\(\displaystyle \text{Area of Shaded Region}=150-\frac{75}{2}\)

Solve.

\(\displaystyle \text{Area of Shaded Region}=\frac{225}{2}\)

Example Question #142 : Triangles

In the figure, a triangle that shares its base with the width of the rectangle has a height that is half the length of the rectangle. Find the area of the shaded region.

8

Possible Answers:

\(\displaystyle 70\)

\(\displaystyle 74\)

\(\displaystyle 78\)

\(\displaystyle 64\)

Correct answer:

\(\displaystyle 78\)

Explanation:

13

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

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 width to find the area.

\(\displaystyle \text{Area of Rectangle}=8 \times 13\)

\(\displaystyle \text{Area of Rectangle}= 104\)

Next, recall how to find the area of a triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times\text{base}\times\text{height}\)

From the question, we know that the height of the triangle is half the length of the rectangle. Use the length of the rectangle to find the height of the triangle:

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

\(\displaystyle \text{height}=\frac{8}{2}\)

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

Since the base of the triangle and the width of the rectangle are the same, we can find the area of the triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times 13\times 4\)

\(\displaystyle \text{Area of Triangle}=26\)

To find the area of the shaded region, subtract the area of the triangle from the area of the rectangle.

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

\(\displaystyle \text{Area of Shaded Region}=104-26\)

Solve.

\(\displaystyle \text{Area of Shaded Region}=78\)

Example Question #23 : How To Find The Area Of An Acute / Obtuse Triangle

In the figure, a triangle that shares its base with the width of the rectangle has a height that is half the length of the rectangle. Find the area of the shaded region.

9

Possible Answers:

\(\displaystyle 300\)

\(\displaystyle 350\)

\(\displaystyle 200\)

\(\displaystyle 250\)

Correct answer:

\(\displaystyle 300\)

Explanation:

13

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

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 width to find the area.

\(\displaystyle \text{Area of Rectangle}=16 \times 25\)

\(\displaystyle \text{Area of Rectangle}= 400\)

Next, recall how to find the area of a triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times\text{base}\times\text{height}\)

From the question, we know that the height of the triangle is half the length of the rectangle. Use the length of the rectangle to find the height of the triangle:

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

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

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

Since the base of the triangle and the width of the rectangle are the same, we can find the area of the triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times 25\times 8\)

\(\displaystyle \text{Area of Triangle}=100\)

To find the area of the shaded region, subtract the area of the triangle from the area of the rectangle.

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

\(\displaystyle \text{Area of Shaded Region}=400-100\)

Solve.

\(\displaystyle \text{Area of Shaded Region}=300\)

Example Question #24 : How To Find The Area Of An Acute / Obtuse Triangle

In the figure, a triangle that shares its base with the width of the rectangle has a height that is half the length of the rectangle. Find the area of the shaded region.

10

Possible Answers:

\(\displaystyle 90\)

\(\displaystyle 100\)

\(\displaystyle 80\)

\(\displaystyle 120\)

Correct answer:

\(\displaystyle 120\)

Explanation:

13

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

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 width to find the area.

\(\displaystyle \text{Area of Rectangle}=8 \times 20\)

\(\displaystyle \text{Area of Rectangle}= 160\)

Next, recall how to find the area of a triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times\text{base}\times\text{height}\)

From the question, we know that the height of the triangle is half the length of the rectangle. Use the length of the rectangle to find the height of the triangle:

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

\(\displaystyle \text{height}=\frac{8}{2}\)

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

Since the base of the triangle and the width of the rectangle are the same, we can find the area of the triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times 20\times 4\)

\(\displaystyle \text{Area of Triangle}=40\)

To find the area of the shaded region, subtract the area of the triangle from the area of the rectangle.

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

\(\displaystyle \text{Area of Shaded Region}=160-40\)

Solve.

\(\displaystyle \text{Area of Shaded Region}=120\)

Example Question #25 : How To Find The Area Of An Acute / Obtuse Triangle

In the figure, a triangle that shares its base with the width of the rectangle has a height that is half the length of the rectangle. Find the area of the shaded region.

11

Possible Answers:

\(\displaystyle 384\)

\(\displaystyle 346\)

\(\displaystyle 378\)

\(\displaystyle 296\)

Correct answer:

\(\displaystyle 378\)

Explanation:

13

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

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 width to find the area.

\(\displaystyle \text{Area of Rectangle}=18 \times 28\)

\(\displaystyle \text{Area of Rectangle}= 504\)

Next, recall how to find the area of a triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times\text{base}\times\text{height}\)

From the question, we know that the height of the triangle is half the length of the rectangle. Use the length of the rectangle to find the height of the triangle:

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

\(\displaystyle \text{height}=\frac{18}{2}\)

\(\displaystyle \text{height}=9\)

Since the base of the triangle and the width of the rectangle are the same, we can find the area of the triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times 28\times 9\)

\(\displaystyle \text{Area of Triangle} =126\)

To find the area of the shaded region, subtract the area of the triangle from the area of the rectangle.

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

\(\displaystyle \text{Area of Shaded Region}=504-126\)

Solve.

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

Example Question #143 : Triangles

In the figure, a triangle that shares its base with the width of the rectangle has a height that is half the length of the rectangle. Find the area of the shaded region.

12

Possible Answers:

\(\displaystyle \frac{246}{4}\)

\(\displaystyle \frac{297}{4}\)

\(\displaystyle \frac{301}{4}\)

\(\displaystyle \frac{287}{4}\)

Correct answer:

\(\displaystyle \frac{297}{4}\)

Explanation:

13

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

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 width to find the area.

\(\displaystyle \text{Area of Rectangle}=9\times 11\)

\(\displaystyle \text{Area of Rectangle}=99\)

Next, recall how to find the area of a triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times\text{base}\times\text{height}\)

From the question, we know that the height of the triangle is half the length of the rectangle. Use the length of the rectangle to find the height of the triangle:

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

\(\displaystyle \text{height}=\frac{9}{2}\)

Since the base of the triangle and the width of the rectangle are the same, we can find the area of the triangle:

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}\times 11\times \frac{9}{2}\)

\(\displaystyle \text{Area of Triangle}=\frac{99}{4}\)

To find the area of the shaded region, subtract the area of the triangle from the area of the rectangle.

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

\(\displaystyle \text{Area of Shaded Region}=99-\frac{99}{4}\)

Solve.

\(\displaystyle \text{Area of Shaded Region}=\frac{297}{4}\)

Example Question #141 : Triangles

A triangle is placed in a parallelogram so they share a base.

1

If the height of the triangle is half that of the parallelogram, find the area of the shaded region.

Possible Answers:

\(\displaystyle 84\)

\(\displaystyle 72\)

\(\displaystyle 48\)

\(\displaystyle 66\)

Correct answer:

\(\displaystyle 72\)

Explanation:

13

In order to find the area of the shaded region, we will need to find the areas of the triangle and of the parallelogram.

First, recall how to find the area of a parallelogram.

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

\(\displaystyle \text{Area of Parallelogram}=12\times8=96\)

Next, recall how to find the area of a triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})\)

Now, find the height of the triangle.

\(\displaystyle \text{Height of Triangle}=\frac{\text{Height of Parallelogram}}{2}\)

\(\displaystyle \text{Height of Triangle}=\frac{8}{2}=4\)

Plug this value in to find the area of the triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(12 \times 4)=24\)

Subtract the two areas to find the area of the shaded region.

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

\(\displaystyle \text{Area of Shaded Region}=96-24=72\)

Example Question #581 : Intermediate Geometry

A triangle is placed in a parallelogram so that they share a base.

2

If the height of the triangle is one-fifth of that of the parallelogram, find the area of the shaded region.

Possible Answers:

\(\displaystyle 134\)

\(\displaystyle 126\)

\(\displaystyle 120\)

\(\displaystyle 118\)

Correct answer:

\(\displaystyle 126\)

Explanation:

13

In order to find the area of the shaded region, we will need to find the areas of the triangle and of the parallelogram.

First, recall how to find the area of a parallelogram.

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

\(\displaystyle \text{Area of Parallelogram}=14\times 10=140\)

Next, recall how to find the area of a triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})\)

Now, find the height of the triangle.

\(\displaystyle \text{Height of Triangle}=\frac{\text{Height of Parallelogram}}{5}\)

\(\displaystyle \text{Height of Triangle}=\frac{10}{5}=2\)

Plug this value in to find the area of the triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(14 \times 2)=14\)

Subtract the two areas to find the area of the shaded region.

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

\(\displaystyle \text{Area of Shaded Region}=140-14=126\)

Example Question #145 : Triangles

A triangle is placed so that it shares a base with a parallelogram.

3

If the height of the triangle is one-third that of the parallelogram, find the area of the shaded region.

Possible Answers:

\(\displaystyle 2.5\)

\(\displaystyle 10.5\)

\(\displaystyle 8.5\)

\(\displaystyle 12.5\)

Correct answer:

\(\displaystyle 12.5\)

Explanation:

13

In order to find the area of the shaded region, we will need to find the areas of the triangle and of the parallelogram.

First, recall how to find the area of a parallelogram.

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

\(\displaystyle \text{Area of Parallelogram}=3\times 5=15\)

Next, recall how to find the area of a triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})\)

Now, find the height of the triangle.

\(\displaystyle \text{Height of Triangle}=\frac{\text{Height of Parallelogram}}{3}\)

\(\displaystyle \text{Height of Triangle}=\frac{3}{3}=1\)

Plug this value in to find the area of the triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(5\times 1)=\frac{5}{2}=2.5\)

Subtract the two areas to find the area of the shaded region.

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

\(\displaystyle \text{Area of Shaded Region}=15-2.5=12.5\)

Example Question #591 : Plane Geometry

A triangle is placed in a parallelogram so that they share a base.

4

If the height of the triangle is half that of the parallelogram, find the area of the shaded region.

Possible Answers:

\(\displaystyle 88\)

\(\displaystyle 92\)

\(\displaystyle 84\)

\(\displaystyle 90\)

Correct answer:

\(\displaystyle 84\)

Explanation:

13

In order to find the area of the shaded region, we will need to find the areas of the triangle and of the parallelogram.

First, recall how to find the area of a parallelogram.

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

\(\displaystyle \text{Area of Parallelogram}=14\times8=112\)

Next, recall how to find the area of a triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})\)

Now, find the height of the triangle.

\(\displaystyle \text{Height of Triangle}=\frac{\text{Height of Parallelogram}}{2}\)

\(\displaystyle \text{Height of Triangle}=\frac{8}{2}=4\)

Plug this value in to find the area of the triangle.

\(\displaystyle \text{Area of Triangle}=\frac{1}{2}(14 \times 4)=28\)

Subtract the two areas to find the area of the shaded region.

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

\(\displaystyle \text{Area of Shaded Region}=112-28=84\)

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