Properties of polygons and circles

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HiSET › Properties of polygons and circles

Questions 1 - 10
1

Pool shape

A quadrilateral is shown, and the angle measures of 3 interior angles are given. Find x, the missing angle measure.

Explanation

The sum of the measures of the interior angles of a quadrilateral is 360 degrees. The sum of the measures of the interior angles of any polygon can be determined using the following formula:

, where is the number of sides.

For example, with a quadrilateral, which has 4 sides, you obtain the following calculation:

Solving for requires setting up an algebraic equation, adding all 4 angles to equal 360 degrees:

Solving for is straightforward: subtract the values of the 3 known angles from both sides:

2

Circle area

The depicted circle has a radius of 3 cm. The arc length between the two points shown on the circle is cm. Find the area of the enclosed sector (highlighted in green).

Explanation

First, find the area and circumference of the circle using the radius and the following formulae for circles:

Substituting in 3 for yields:

Next, find what fraction of the total circumference is between the two points on the circle (the arc length).

Finally, use this fraction to calculate the area of the enclosed sector. Note that this area is proportional to the above fraction. In other words:

So the Sector Area is one sixth of the total area.

Cross multiply:

Divide both sides by 6, then simplify to get the final answer:

3

Circle area

The depicted circle has a radius of 3 cm. The arc length between the two points shown on the circle is cm. Find the area of the enclosed sector (highlighted in green).

Explanation

First, find the area and circumference of the circle using the radius and the following formulae for circles:

Substituting in 3 for yields:

Next, find what fraction of the total circumference is between the two points on the circle (the arc length).

Finally, use this fraction to calculate the area of the enclosed sector. Note that this area is proportional to the above fraction. In other words:

So the Sector Area is one sixth of the total area.

Cross multiply:

Divide both sides by 6, then simplify to get the final answer:

4

Pool shape

A quadrilateral is shown, and the angle measures of 3 interior angles are given. Find x, the missing angle measure.

Explanation

The sum of the measures of the interior angles of a quadrilateral is 360 degrees. The sum of the measures of the interior angles of any polygon can be determined using the following formula:

, where is the number of sides.

For example, with a quadrilateral, which has 4 sides, you obtain the following calculation:

Solving for requires setting up an algebraic equation, adding all 4 angles to equal 360 degrees:

Solving for is straightforward: subtract the values of the 3 known angles from both sides:

5

Give the perimeter of a regular octagon in yards if the length of each side is feet.

yards

yards

yards

yards

yards

Explanation

The perimeter of a regular octagon - the sum of the lengths of its (eight congruent) sides - is eight times the common sidelength, so the perimeter of the octagon is feet. One yard is equivalent to three feet, so divide this by conversion factor 3 to get yards - the correct response.

6

Give the perimeter of a regular octagon in yards if the length of each side is feet.

yards

yards

yards

yards

yards

Explanation

The perimeter of a regular octagon - the sum of the lengths of its (eight congruent) sides - is eight times the common sidelength, so the perimeter of the octagon is feet. One yard is equivalent to three feet, so divide this by conversion factor 3 to get yards - the correct response.

7

Intercepted

Refer to the above figure. Give the ratio of the area of Sector 2 to that of Sector 1.

Explanation

The ratio of the area of the larger Sector 2 to that of smaller Sector 1 is equal to the ratio of their respective arc measures - that is,

.

Therefore, it is sufficient to find these arc measures.

If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Add 360 to both sides:

Divide both sides by 2:

,

the degree measure of .

It follows that

By the Arc Addition Principle,

Since , the central angle which intercepts , is a right angle, . By substitution,

,

and

The ratio is equal to

,

a 5 to 1 ratio.

8

and are complementary angles.

and are supplementary angles.

Evaluate .

Explanation

and are supplementary angles, so, by definition,

, so substitute and solve for :

and are complementary angles, so, by definition,

Substitute and solve for :

- that is, the angles have the same measure. Therefore,

.

9

Intercepted 2

is the center of the above circle. Calculate .

Explanation

is the central angle that intercepts , so

.

Therefore, we need to find to obtain our answer.

If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Subtract 360 from both sides:

Divide both sides by :

,

the measure of and, consequently, that of .

10

Intercepted 2

is the center of the above circle, and . Evaluate the length of .

Explanation

The radius of the circle is given to be . The total circumference of the circle is times this, or

.

The length of is equal to

Thus, it is first necessary to find the degree measure of . If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Subtract 360 from both sides:

Divide both sides by :

,

the degree measure of .

Thus, the length of is

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