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Intermediate Geometry : Plane Geometry

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Example Questions

Example Question #1 : How To Find The Perimeter Of A Pentagon

If the side of a pentagon is $5\sqrt5$ , what is the perimeter?

Possible Answers:

125

$5+5\sqrt5$

$25\sqrt5$

$5\sqrt5$

$10\sqrt5$

Correct answer:

$25\sqrt5$

Explanation:

Write the perimeter formula for pentagons and substitute the side length.

P=5s

$P=5(5\sqrt5)=25\sqrt5$

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Example Question #2 : How To Find The Perimeter Of A Pentagon

Determine the perimeter of a pentagon if the side length is $\frac{\sqrt2}{10}$ .

Possible Answers:

$\frac{\sqrt2}{2}$

$\sqrt2$

$\frac{\sqrt10}{10}$

$\frac{1}{2}$

Correct answer:

$\frac{\sqrt2}{2}$

Explanation:

Write the formula to find the perimeter for a pentagon.

P=5s

Substitute the side into the perimeter formula.

$P=5(\frac{\sqrt2}{10})=\frac{\sqrt2}{2}$

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Example Question #5 : How To Find The Perimeter Of A Pentagon

In the figure below, if the length of is one-third the length of , what is the perimeter of the pentagon ABCDE ?

Possible Answers:

56.99

78.08

65.65

60.33

Correct answer:

65.65

Explanation:

Notice that the pentagon is made up of one rectangle and two right triangles. Start by finding the lengths of and .

Let be the length of . Then, it follows that is the length of .

From the figure, we know that AF+FD=AD . Since we are working with a rectangle, AD=BC=24 .

Now, solve for .

x+3x=24

4x=24

x=6

So then, $AF=6 \text{ and }FD=18$ .

Now that we know AF=6 , we can use the Pythagorean theorem to solve for the length of .

$AE=\sqrt{(AF)^2+(FE)^2}$

Plug in the lengths of the legs to find the length of the hypotenuse.

$AE=\sqrt{6^2+4^2}=\sqrt{52}$

Next, find the length of by using the Pythagorean theorem.

$DE=\sqrt{(FD)^2+(EF)^2}$

$DE=\sqrt{18^2+4^2}=\sqrt{340}$

Finally, add up all the side lengths of the pentagon to find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=8+24+8+\sqrt{340}+\sqrt{52}=65.65$

Make sure to round to places after the decimal.

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Example Question #6 : How To Find The Perimeter Of A Pentagon

In pentagon ABCDE below, the length of is one-third that of . Find the perimeter of the pentagon.

Possible Answers:

146.41

128.57

133.86

152.09

Correct answer:

146.41

Explanation:

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 38=\frac{9}{AF}$

$AF=\frac{9}{\tan38}=11.52$

Now, we know that is three times the length of .

FD=3(AF)=3(11.52)=34.56

Since we have rectangle ABCD , AD=BC .

Thus, AF+FD=AD=BC

BC=11.52+34.56=46.08

Now, we can use cosine to find the length of .

$\sin38=\frac{9}{AE}$

$AE=\frac{9}{\sin38}=14.62$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{9^2+34.56^2}=35.71$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=25+46.08+25+35.71+14.62=146.41$

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Example Question #7 : How To Find The Perimeter Of A Pentagon

In pentagon ABCDE below, the length of is one-third the length of . Find the perimeter of the pentagon.

Possible Answers:

271.58

273.26

280.69

263.41

Correct answer:

273.26

Explanation:

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 26=\frac{13}{AF}$

$AF=\frac{13}{\tan 26}=26.65$

Now, we know that is three times the length of .

FD=3(AF)=3(26.65)=79.95

Since we have rectangle ABCD , AD=BC .

Thus, AF+FD=AD=BC

BC=26.65+79.95=106.60

Now, we can use sine to find the length of .

$\sin 26=\frac{13}{AE}$

$AE=\frac{13}{\sin 26}=29.66$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{13^2+79.95^2}=81.00$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=28+106.60+28+81.00+29.66=273.26$

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Example Question #1 : How To Find The Length Of The Diagonal Of A Pentagon

In the pentagon below, the length of is one-third that of . Find the length of diagonal .

Possible Answers:

85.99

80.50

81.72

79.38

Correct answer:

80.50

Explanation:

Start by using the Pythagorean theorem to find the length of .

$AF=\sqrt{(AE)^2-(EF)^2}$

$AF=\sqrt{22^2-18^2}=\sqrt{324}$

Since is one-third the length of , we know that the length of must be $3\sqrt{340}$ .

Now, notice that the pentagon is made up of one rectangle and two right triangles.

Focus on rectangle ABCD . We can find the length of by adding together the lengths of and .

$\text{AD}=\sqrt{324}+3\sqrt{324}=4\sqrt{324}$

Since we are working with a rectangle, we know that AD=BC . Thus, we can use the Pythagorean Theorem again to find the length of .

$AC=\sqrt{(AB)^2+(BC)^2}$

$AC=\sqrt{36^2+(4\sqrt{324})^2}=\sqrt{6480}=80.50$

Make sure to round to two places after the decimal.

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Example Question #1 : How To Find The Length Of The Diagonal Of A Pentagon

What is the length of a diagonal of a pentagon with side length of 20 centimeters (to the nearest tenth)?

Possible Answers:

$34.4\;cm$

$30.0\;cm$

$32.4\;cm$

Correct answer:

$32.4\;cm$

Explanation:

Let be the length fo a diagonal.

One interior angle of a pentagon measures

$C=\frac{180(5-2)}{5}=108^{\circ }$

A diagonal of this pentagon, along with two sides, forms an isosceles triangle with sides 20, 20 and ; the side of length is opposite the angle of measure $108^{\circ }$ , and each side of length 20 is opposite an angle of measure $\frac{180-108}{2}=36^{\circ }$ .

The Law of Sines helps us to determine :

$\frac{d }{sin\;108^{\circ}}=\frac{20 }{sin\;36^{\circ}}$

$d=\frac{20\cdot sin\;108^{\circ} }{sin\;36^{\circ}}\approx\frac{20\cdot 0.9511 }{0.5878}\approx 32.4$

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Example Question #753 : Plane Geometry

A regular pentagon has a perimeter of $\small 125$ . Find the length of one side of the pentagon.

Possible Answers:

$\small 35$

$\small 25$

$\small 20$

$\small 75$

$\small 15$

Correct answer:

$\small 25$

Explanation:

By definition a regular pentagon must have five equivalent sides and five equivalent interior angles. Since this problem provides the measurement for the total perimeter of the pentagon, work backwards using the formula:

$\small p=5(s)$ , where $\small s =$ the length of one side of the pentagon.

$\small 125=5(s)$

$\small s=\frac{125}{5}=25$

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Example Question #1 : How To Find The Length Of The Side Of A Pentagon

A regular pentagon has a perimeter of $\small \small 75$ . Find the length of one side of the pentagon.

Possible Answers:

$\small 20$

$\small 5$

$\small 12$

$\small 25$

$\small 15$

Correct answer:

$\small 15$

Explanation:

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Example Question #1 : How To Find The Length Of The Side Of A Pentagon

A regular pentagon has an area of $\small 80$ square units and an apothem measurement of $\small 4.7$ . Find the length of one side of the pentagon.

Possible Answers:

$\small 16$

$\small 4.3$

$\small 6.8$

$\small 4.2$

$\small 8.6$

Correct answer:

$\small 6.8$

Explanation:

A regular pentagon must have five equivalent sides and five equivalent interior angles. Regular pentagons can be divided up into five equivalent interior triangles, where the base of the triangle is a side length and the height of the triangle is the apothem. This problem provides the total area for the pentagon.

Therefore, you must work backwards using the area formula: $\small a=\frac{base\times height}{2}$ , where the base is equal to the length of one side of the triangle and the height of the triangle is the measurement of the apothem. However, this will only provide the measurement for one of the five interior triangles, thus you first need to divide the total area by $\small 5.$

The solution is:

$\small a=80$

So, area of one of the five interior triangles is equal to: $\small 80\div5=16$

Now, apply the area formula:

$\small 16=\frac{base\times 4.7}{2}$

$\small 32=base\times4.7$

$\small base=32\div4.7=6.8$

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Perimeter Of A Pentagon

Example Question #2 : How To Find The Perimeter Of A Pentagon

Example Question #5 : How To Find The Perimeter Of A Pentagon

Example Question #6 : How To Find The Perimeter Of A Pentagon

Example Question #7 : How To Find The Perimeter Of A Pentagon

Example Question #1 : How To Find The Length Of The Diagonal Of A Pentagon

Example Question #1 : How To Find The Length Of The Diagonal Of A Pentagon

Example Question #753 : Plane Geometry

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

All Intermediate Geometry Resources

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