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ACT Math : Plane Geometry

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Example Questions

Example Question #601 : Plane Geometry

Polygon ABCDEFG is a regular seven-sided polygon, or heptagon, with perimeter 500. Which choice comes closest to the length of diagonal $\overline{AD}$ ?

Possible Answers:

140

170

180

150

160

Correct answer:

160

Explanation:

Congruent sides $\overline{AB}$ , $\overline{BC}$ , and $\overline{CD}$ , and the diagonal $\overline{AD}$ form an isosceles trapezoid.

$\angle ABC$ and $\angle BCD$ . being angles of a seven-sided regular polygon, have measure

$m \angle ABC = m \angle BCD = \frac{180 ^{\circ } (7-2)}{7} \approx 128.57^{\circ }$

The other two angles are supplementary to these:

$m \angle DAB = m \angle ADC \approx 180 ^{\circ }- 128.57^{\circ }\approx 51.43 ^{\circ }$

The length of one side is one-seventh of 500, so

$AB = BC =CD= \frac{500}{7} \approx 71.42$

The trapezoid formed is below (figure NOT drawn to scale):

Thingy

Altitudes $\overline{BM}$ and $\overline{CN}$ to the base have been drawn, so

AD = AM + MN + ND

AD = AM + BC+ AM

$AD = 2 \cdot AM + 71.42$

$\frac{AM}{AB} = \cos 51.43^{ \circ}$

$AM=AB \cos 51.43^{ \circ} \approx 71.42 \cdot 0.6235 \approx 44.53$

$AD \approx 2 \cdot 44.53 + 71.42 \approx 160.48$

This makes 160 the best choice.

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

Polygon ABCDEFG is a regular seven-sided polygon, or heptagon, with perimeter 500. Which choice comes closest to the length of diagonal $\overline{AC}$ ?

Possible Answers:

140

110

100

120

130

Correct answer:

130

Explanation:

Congruent sides $\overline{AB}$ and $\overline{BC}$ and the diagonal $\overline{AC}$ form an isosceles triangle.

$\angle ABC$ , being an angle of a seven-sided regular polygon, has measure

$m \angle ABC = \frac{180 ^{\circ } (7-2)}{7} \approx 128.57^{\circ }$

The other two are congruent, and each has measure

$m \angle CAB = m \angle ACB = \frac{180^{\circ }- 128.57^{\circ }}{2}\approx 25.71^{\circ}$

The length of one side is one-seventh of 500, or

$AB = BC = \frac{500}{7} \approx 71.42$

can be found using the Law of Sines:

$\frac{AC}{\sin m \angle B} = \frac{AB}{\sin m \angle ACB}$

$\frac{AC}{\sin 128.57^{\circ} } = \frac{71.42}{\sin 25.71^{\circ}}$

$AC \approx \frac{71.42}{\sin 25.71^{\circ}}\cdot \sin 128.57^{\circ}$

$AC \approx \frac{71.42}{0.4339}\cdot 0.7818$

$AC \approx 128.7$

Of the given choices, 130 comes closest.

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Example Question #1 : Other Polygons

Polygon ABCDEFGHI is a regular nine-sided polygon, or nonagon, with perimeter 500. Which choice comes closest to the length of diagonal $\overline{AC}$ ?

Possible Answers:

100

110

105

Correct answer:

105

Explanation:

Congruent sides $\overline{AB}$ and $\overline{BC}$ and the diagonal $\overline{AC}$ form an isosceles triangle.

$\angle ABC$ , being an angle of a nine-sided regular polygon, has measure

$m \angle ABC = \frac{180 ^{\circ } (9-2)}{9} =140^{\circ }$

The other two are congruent, and each has measure

$m \angle CAB = m \angle ACB = \frac{180^{\circ }- 140^{\circ }}{2}\approx 20^{\circ}$

The length of one side is one-ninth of 500, or

$AB = BC = \frac{500}{9} \approx 55.56$

can be found using the Law of Sines:

$\frac{AC}{\sin m \angle B} = \frac{AB}{\sin m \angle ACB}$

$\frac{AC}{\sin 140^{\circ} } = \frac{55.56}{\sin 20^{\circ}}$

$AC \approx \frac{55.56}{\sin 20^{\circ}}\cdot \sin 140^{\circ}$

$AC \approx \frac{55.56}{0.3420}\cdot 0.6428$

$AC \approx 104.4$

Of the given choices, 105 comes closest.

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Example Question #2 : Other Polygons

What is the side length of a 12-sided polygon with a perimeter of 48 cm?

Possible Answers:

2cm

26cm

12cm

4cm

576cm

Correct answer:

4cm

Explanation:

To find the side length, , of an -sided polygon with a perimeter of .

Use the formula:
$\\s = p/n \newline s= 48/12 \\s= 4cm$

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Example Question #601 : Plane Geometry

In the diagram below, what is $\theta$ equal to?

Hexagon_sides

Possible Answers:

$\sin^{-1}\left ( \frac{r}{s} \right )$

$\tan^{-1}\left ( \frac{r}{s} \right )$

$\tan^{-1}\left ( \frac{s}{r} \right )$

$\tan \left ( \frac{r}{s} \right )$

$\tan \left ( \frac{s}{r} \right )$

Correct answer:

$\tan^{-1}\left ( \frac{s}{r} \right )$

Explanation:

The figure given is a hexagon with an embedded triangle. The fact that it is embedded in a triangle is mainly to throw you off, as it has little to no consequence on the correct answer. Of the available answer choices, you must choose a relationship that would give the value of $\theta$ . Tangent describes the relationship between an angle and the opposite and adjacent sides of that angle. Or in other words, tan $\theta$ = opposite side/adjacent side. However, when solving for an angle, we must use the inverse function. Therefore, if we know the opposite and adjacent sides are, we can use the inverse of the tangent, or arctangent (tan^-1), of $\frac{s}{r}$ to find $\theta$ .

Thus,

$\theta =\tan^{-1}\left ( \frac{s}{r} \right )$

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Example Question #1 : Other Polygons

Shape_1

What is the value of angle in the figure above?

Possible Answers:

$110^{\circ}$

$75^{\circ}$

$50^{\circ}$

$35^{\circ}$

$100^{\circ}$

Correct answer:

$100^{\circ}$

Explanation:

Begin by noticing that the upper-right angle of this figure is supplementary to $70^{\circ}$ . This means that it is $180^{\circ}-70^{\circ}=110^{\circ}$ :

Shape_1

Now, a quadrilateral has a total of $360^{\circ}$ . This is computed by the formula degrees = 180 * (s-2) , where represents the number of sides. Thus, we know:

$110^{\circ}+70^{\circ}+80^{\circ}+x=360^{\circ}$

This is the same as $260^{\circ} + x = 360^{\circ}$

Solving for , we get:

$x=100^{\circ}$

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Example Question #603 : Plane Geometry

Fig_2

What is the angle measure for the largest unknown angle in the figure above? Round to the nearest hundredth.

Possible Answers:

$186.36^{\circ}$

$84.14^{\circ}$

$140.14^{\circ}$

$74.55^{\circ}$

$149.1^{\circ}$

Correct answer:

$186.36^{\circ}$

Explanation:

The total degree measure of a given figure is given by the equation 180*(s-2) , where represents the number of sides in the figure. For this figure, it is: $180*(6-2)=180*4 = 720^{\circ}$

Therefore, we know that the sum of the angles must equal $720^{\circ}$ . This gives us the equation:

$90^{\circ}+120^{\circ}+100^{\circ}+x+2.5x+2x=720^{\circ}$

Simplifying, this is:

$310^{\circ}+5.5x=720^{\circ}$

Now, just solve for :

5.5x=410

$x=\frac{410}{5.5}$

The largest of the unknown angles is 2.5x or $2.5 * \frac{410}{5.5}=186.36363636363636...^{\circ}$

Rounding, this is $186.36^{\circ}$ .

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Example Question #4 : How To Find An Angle In A Polygon

What is the interior angle of a $\textup{9-sided}$ polygon (a nonagon)? Round answer to the nearest hundredth if necessary.

Possible Answers:

$140^{\circ}$

$147.27^{\circ}$

$1260^{\circ}$

$128.57^{\circ}$

$360^{\circ}$

Correct answer:

$140^{\circ}$

Explanation:

To find the interior angle of an sided polygon, first find the total number of degrees in the polygon by the formula:
$(n-2)*180^{\circ}$ . For us that yields:
$(9-2)*180^{\circ} = 1260^{\circ}$ . Next we divide the total number of degrees by the number of sides:

$1260^{\circ}/9 = 140^{\circ}$

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Example Question #12 : Other Polygons

What is the total number of degrees in a $\textup{12-sided}$ polygon?

Possible Answers:

$2160^{\circ}$

$1800^{\circ}$

$144^{\circ}$

$150^{\circ}$

$1440^{\circ}$

Correct answer:

$1800^{\circ}$

Explanation:

To find the total number of degrees in an -sided polygon, use the formula:

(n-2)*180 thus we see that (12-2)*180 = $1800^{\circ}$

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Example Question #5 : How To Find An Angle In A Polygon

What is the interior angle of a $\textup{16-sided}$ polygon?

Possible Answers:

$360^{\circ}$

$2520^{\circ}$

$154.29^{\circ}$

$157.5^{\circ}$

$160^{\circ}$

Correct answer:

$157.5^{\circ}$

Explanation:

To find the interior angle of a regular, -sided polygon, use the formula:

((n-2)*180)/n :

Thus we see that $(16-2)*180^{\circ} = 2520^{\circ}$ and

$2520^{\circ}/16 = 157.5^{\circ}$

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #601 : Plane Geometry

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

Example Question #1 : Other Polygons

Example Question #2 : Other Polygons

Example Question #601 : Plane Geometry

Example Question #1 : Other Polygons

Example Question #603 : Plane Geometry

Example Question #4 : How To Find An Angle In A Polygon

Example Question #12 : Other Polygons

Example Question #5 : How To Find An Angle In A Polygon

All ACT Math Resources

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