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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #2 : How To Find If Parallelograms Are Similar

A parallelogram has adjacent sides with the lengths of and . Find a pair of possible adjacent side lengths for a similar parallelogram.

Possible Answers:

1:4

1:3

3:1

4:1

Correct answer:

3:1

Explanation:

Since the two parallelogram are similar, each of the corresponding sides must have the same ratio.

The ratio of the first parallelogram is:

$15:5\rightarrow 3:1$

Applying this ratio we are able to find the lengths of the second parallelogram.

$15:5=(15\div 5):(5\div 5)=3:1$

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Example Question #1 : How To Find If Parallelograms Are Similar

A parallelogram has adjacent sides with the lengths of and . Find a pair of possible adjacent side lengths for a similar parallelogram.

Possible Answers:

4:1

7:2

2:9

16:4

Correct answer:

7:2

Explanation:

Since the two parallelogram are similar, each of the corresponding sides must have the same ratio.

The ratio of the first parallelogram is:

$28:8 \rightarrow 28\div4:8\div4 \rightarrow 7:2$

Thus by simplifying the ratio we can see the lengths of the similar triangle.

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Example Question #1 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a base of $12\ cm$ and an area of $42\ cm^{2}$ . What must be the height of this triangle?

Possible Answers:

$9\ cm$

$10\ cm$

$6\ cm$

$8\ cm$

$7\ cm$

Correct answer:

$7\ cm$

Explanation:

$A=\frac{1}{2}bh$ .

$6x=42$

$x=7$

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Example Question #1 : Triangles

An isosceles triangle has a perimeter of . If the base of the triangle is two less than two times the length of each leg, what is the height of the triangle?

Possible Answers:

3.32

4.82

The height of the triangle cannot be determined with the given information.

3.11

Correct answer:

3.32

Explanation:

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be 2x-2 .

Use the information given about the perimeter to solve for .

x+x+2x-2=22

4x-2=22

4x=24

x=6

Plug this value in to find the length of the base.

$\text{Length of Base}=2(6)-2=10$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-5^2}=\sqrt{11}=3.32$

Make sure to round to places after the decimal.

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Example Question #1 : How To Find The Height Of Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle has a perimeter of . If the base of the triangle is less than three times the length of a leg, what is the height of the triangle?

Possible Answers:

Correct answer:

Explanation:

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be 3x-18 .

Use the information given about the perimeter to solve for .

x+x+3x-18=32

5x-18=32

5x=50

x=10

Plug this value in to find the length of the base.

$\text{Length of Base}=3(10)-18=12$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{10^2-6^2}=\sqrt{64}=8$

Make sure to round to places after the decimal.

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Example Question #3 : Triangles

An isosceles triangle has a perimeter of . If the length of the base is one less than twice the length of a leg, what is the height of the triangle?

Possible Answers:

2.90

1.29

2.44

2.18

Correct answer:

2.18

Explanation:

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be 2x-1 .

Use the information given about the perimeter to solve for .

x+x+2x-1=19

4x-1=19

4x=20

x=5

Plug this value in to find the length of the base.

$\text{Length of Base}=2(5)-1=9$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{5^2-4.5^2}=\sqrt{4.75}=2.18$

Make sure to round to places after the decimal.

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Example Question #1 : Triangles

An isosceles triangle has a perimeter of . If the length of the base is fourteen less than three times the length of a leg, what is the height of the triangle?

Possible Answers:

4.93

5.66

5.89

5.09

Correct answer:

5.66

Explanation:

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be 3x-14 .

Use the information given about the perimeter to solve for .

x+x+3x-14=16

5x-14=16

5x=30

x=6

Plug this value in to find the length of the base.

$\text{Length of Base}=3(6)-14=4$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-2^2}=\sqrt{32}=5.66$

Make sure to round to places after the decimal.

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Example Question #1 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a perimeter of . If the length of the base is two less than one-third of the length of a leg, what is the height of the triangle?

Possible Answers:

8.19

8.99

8.82

9.02

Correct answer:

8.99

Explanation:

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{3}x-2$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{3}x-2=19$

$\frac{7}{3}x-2=19$

$\frac{7}{3}x=21$

x=9

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{3}(9)-2=1$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{9^2-(\frac{1}{2})^2}=\sqrt{80.75}=8.99$

Make sure to round to places after the decimal.

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Example Question #6 : Triangles

An isosceles triangle has a perimeter of . If the length of the base is ten less than twice the length of a leg, what is the height of the triangle?

Possible Answers:

9.87

10.12

9.75

8.90

Correct answer:

9.75

Explanation:

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be 2x-10 .

Use the information given about the perimeter to solve for .

x+x+2x-10=38

4x-10=38

4x=48

x=12

Plug this value in to find the length of the base.

$\text{Length of Base}=2(12)-10=14$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{12^2-7^2}=\sqrt{95}=9.75$

Make sure to round to places after the decimal.

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Example Question #7 : Triangles

An isosceles triangle has a perimeter of . If the length of the base is four more than one-third of the length of a leg, what is the height of the triangle?

Possible Answers:

1.97

1.20

1.59

1.66

Correct answer:

1.66

Explanation:

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{3}x+4$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{3}x+4=11$

$\frac{7}{3}x+4=11$

$\frac{7}{3}x=7$

x=3

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{3}(3)+4=5$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{3^2-(\frac{5}{2})^2}=\sqrt{2.75}=1.66$

Make sure to round to places after the decimal.

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #2 : How To Find If Parallelograms Are Similar

Example Question #1 : How To Find If Parallelograms Are Similar

Example Question #1 : Acute / Obtuse Isosceles Triangles

Example Question #1 : Triangles

Example Question #1 : How To Find The Height Of Of An Acute / Obtuse Isosceles Triangle

Example Question #3 : Triangles

Example Question #1 : Triangles

Example Question #1 : Acute / Obtuse Isosceles Triangles

Example Question #6 : Triangles

Example Question #7 : Triangles

All Intermediate Geometry Resources

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