ACT Math : Other Polygons

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #1 : Plane Geometry

Polygon

All segments of the polygon meet at right angles (90 degrees). The length of segment \overline{AB} is 10. The length of segment \overline{BC} is 8. The length of segment \overline{DE} is 3. The length of segment \overline{GH} is 2.

Find the perimeter of the polygon.

Possible Answers:

\dpi{100} \small 44

\dpi{100} \small 48

\dpi{100} \small 42

\dpi{100} \small 40

\dpi{100} \small 46

Correct answer:

\dpi{100} \small 46

Explanation:

The perimeter of the polygon is 46. Think of this polygon as a rectangle with two of its corners "flipped" inwards. This "flipping" changes the area of the rectangle, but not its perimeter; therefore, the top and bottom sides of the original rectangle would be 12 units long \dpi{100} \small (10+2=12). The left and right sides would be 11 units long \dpi{100} \small (8+3=11). Adding all four sides, we find that the perimeter of the recangle (and therefore, of this polygon) is 46.

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

In the figure below, each pair of intersecting line segments forms a right angle, and all the lengths are given in feet. What is the perimeter, in feet, of the figure?

 

Quad

Possible Answers:

Correct answer:

Explanation:

Fill in the missing sides by thinking about the entire figure as a big rectangle. In the figure below, the large rectangle is outlined in blue and the missing numbers are supplied in red.

 

Quad

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

What is the perimeter of a regular,  polygon with a side length of ?

Possible Answers:

Correct answer:

Explanation:

To find the perimeter of an -sided polygon with a side length of , simply multiply the side length by the number of sides:

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

Polygon  is a regular nine-sided polygon, or nonagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Possible Answers:

Correct answer:

Explanation:

Congruent sides , and , and the diagonal  form an isosceles trapezoid. 

 and . being angles of a nine-sided regular polygon, have measure

The other two angles are supplementary to these:

The length of one side of the nonagon is one-ninth of 500, so

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

Thingy

Altitudes  and  to the base have been drawn, so

This makes 140 the best choice.

Example Question #2 : How To Find The Length Of A Diagonal Of A Polygon

Polygon  is a regular seven-sided polygon, or heptagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Possible Answers:

Correct answer:

Explanation:

Congruent sides , and , and the diagonal  form an isosceles trapezoid. 

 and . being angles of a seven-sided regular polygon, have measure

The other two angles are supplementary to these:

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

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

Thingy

Altitudes  and  to the base have been drawn, so

This makes 160 the best choice.

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

Polygon  is a regular seven-sided polygon, or heptagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Possible Answers:

Correct answer:

Explanation:

Congruent sides  and  and the diagonal  form an isosceles triangle. 

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

The other two are congruent, and each has measure

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

 can be found using the Law of Sines:

Of the given choices, 130 comes closest.

Example Question #2 : How To Find The Length Of A Side Of A Polygon

Polygon  is a regular nine-sided polygon, or nonagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Possible Answers:

Correct answer:

Explanation:

Congruent sides  and  and the diagonal  form an isosceles triangle. 

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

The other two are congruent, and each has measure

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

 can be found using the Law of Sines:

Of the given choices, 105 comes closest.

Example Question #1 : Other Polygons

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

Possible Answers:

Correct answer:

Explanation:

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

Use the formula: 

Example Question #1 : How To Find An Angle In A Polygon

In the diagram below, what is  equal to?

Hexagon_sides

Possible Answers:

Correct answer:

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 Tangent describes the relationship between an angle and the opposite and adjacent sides of that angle. Or in other words, tan = 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  to find .

Thus,

 

 

Example Question #1 : How To Find An Angle In A Polygon

Shape_1

What is the value of angle  in the figure above?

Possible Answers:

Correct answer:

Explanation:

Begin by noticing that the upper-right angle of this figure is supplementary to .  This means that it is :

Shape_1

Now, a quadrilateral has a total of . This is computed by the formula , where  represents the number of sides. Thus, we know:

This is the same as 

Solving for , we get:

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