Park Forest Math Team

Meet #4

GeometryGeometry

Self-study Packet

Problem Categories for this Meet: 1. Mystery: Problem solving

2. Geometry: Angle measures in plane figures including supplements and complements

3. Number Theory: Divisibility rules, factors, primes, composites

4. Arithmetic: Order of operations; mean, median, mode; rounding; statistics

5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown including identities

Important information you need to know about GEOMETRY: Properties of Circles

Area of a Circle: A=!r2 Circumference of a Circle: C=!d or C=2!r *(Very Important!) Be sure to use the test’s given value for ! and not the ! button on your calculator!!! Other properties: • If AC is a diameter of a circle and point B is any other point on the circle,

angle ABC will be a right angle.

• The measure of an inscribed angle is half the measure of the arc it “subtends.” For example, if arc AB is 70°, then the measure of angle ADB is 35°

Category 2 50th anniversary edition50th anniversary edition50th anniversary edition50th anniversary editionGeometry Meet #4 - February, 2014 Calculator Meet 1) Angle ABC is a right angle. AB is the diameter of a semi-circle. How many square cm are in the area of the entire figure if AC = 51 cm and BC = 45 cm? Use . 2) The circumference of the larger circle is 46π . The circumference of the smaller circle is 14π . What fractional part of the larger circle is shaded? Express your answer as a common fraction (lowest terms). 3) Arc VWU is a quarter-circle with point Y at the center. XZ = 10 cm. The sum of the length and width of rectangle WXYZ is 13 cm. How many centimeters are in the perimeter of the figure bounded by the points VWUZXV? Use 3.14π ≈ . ANSWERS 1) __________ sq. cm 2) __________ 3) __________ cm

B

A

C

X W

U Z

Y

V

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Solutions to Category 2 Geometry Meet #4 - February, 2014 Answers 1) Use the Pythagorean Theorem to find the length of diameter AB: 1) 763.2

2) 480529

3) 32.7 So, the radius of AB is half of 24, or 12. The total area of the entire figure = (area of semi-circle) + (area of triangle) 2) Using the formula for the circumference of a circle, C=2 rπ , we find that the radius of the smaller circle is 7 and the radius of the larger circle is 23. Using the formula for the area of a circle, 2rπ , we find that the area of the smaller circle is 49π and the area of the larger circle is 529π . By subtracting the area of the smaller circle from the area of the larger circle, we get the area of the shaded region = 480π . Therefore, the fractional part of the larger circle that is shaded is

480529

ππ = 480

529.

3) The key that unlocks this puzzle is the notion that the diagonals of a rectangle are congruent, so that XZ = YW = 10 cm = the radius of the circle = YU = VY. The length of the arc VWU, the quarter-circle, is 0.25(2)(3.14)(10), or 15.7 cm. VX + XY = radius = 10 and YZ + ZU = radius = 10. VX + XY + YZ + ZU = 10 + 10 = 20 VX + (length of rectangle + width of rectangle) + ZU = 20 VX + (13) + ZU = 20, therefore VX + ZU = 7. (Tricky, huh!!) So, perimeter of VWUZXV = 7 + 10 + 15.7 = 32.7 cm.

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Meet #4 February 2012 Calculators allowed

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Category 2 – Geometry

Use

1. The area of a circle is square inches.

How many inches are there in its circumference?

Give an exact answer with no rounding.

2. The radius of the circle shown is centimeters.

The radius points to o‘clock, and the radius

points to o’clock.

How many centimeters are there in the arc ?

3. A square is inscribed inside a circle.

What percentage of the square’s perimeter is the circle’s circumference?

Round your answer to the nearest whole percent.

Answers

1. ___________inches

2. ___________cm

3. ____________ %

B

A

O

Meet #4 February 2012 Calculators allowed

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Solutions to Category 2 – Geometery

1. The area of a circle is so in our case

the radius is √

√ inches.

The circumference is inches.

2. The hours on the clock divide the central angle to equal parts (each one

being degrees then). The arc from to will measure

of the whole

circumference, or in our case

cm

3. If we call the square’s side , then its perimeter is .

Its diagonal is √ , and that is the circle’s diameter, so the circle’s

circumference equals √

Answers

4.

5.

6.

You may use a calculator today!

Category 2 - Geometry

Meet #4, February 2010

1. Assume that the Earth orbits the Sun along a perfect circular orbit with a radius of

150 million kilometers, and completes the orbit in 365 days and 6 hours.

What is the Earth’s average speed around the Sun?

Express your answer in Kilometers per Hour (km/hour), rounded to the nearest

integer. Use 𝜋 = 3.14

2. Kite ABCD is inscribed inside a circle whose center is point O.

∠BDC=25 degrees.

How many degrees are in the measure of ∠BOD?

[A Kite is made up of two isosceles triangles]

3. The radii (plural of radius) of both circles in the diagram measure 10 inches.

They intersect each other in such a way that the distance AB measures 10 inches.

How many inches are in the perimeter of the resulting shape?

Express your answer in inches, rounded to the nearest hundredth. Use 𝜋 = 3.141592.

Answers

1. _______________

2. _______________

3. _______________

B

A

D

C

B

A O

You may use a calculator today!

Solutions to Category 2 - Geometry

Meet #4, February 2010

1. Remember that Speed = Distance / Time.

The distance, in kilometers, is the perimeter of the orbit, namely:

2 ∙ 𝜋 ∙ 𝑅 = 2 ∙ 3.14 ∙ 150 ∙ 106 kilometers. The time, 365 days plus 6 hours, equals to

365 × 24 + 6 hours. The average speed then is:

2 ∙ 3.14 ∙ 150 ∙ 106

(365 × 24 + 6)=

9.42 ∙ 108

8,766= 107,460.64… ≅ 107,461 𝑘𝑚/𝑕𝑜𝑢𝑟

2. ∠𝐵𝐷𝐶 = 25 = ∠𝐷𝐵𝐶 Therefore ∠𝐵𝐶𝐷 = 130 degrees (to complete to 180 degrees).

∠𝐵𝐴𝐷 is inscribed on the chord BD, and so equals 180 − ∠𝐵𝐶𝐷 = 50 degrees.

Finally, ∠𝐵𝑂𝐷 = 2 × ∠𝐵𝐴𝐷 = 100 degrees, since it’s the central angle on the same

chord. Something to think about: Does ABCD have to be a kite?

3. The perimeter is the sum of the two circles’ perimeters, minus the two arcs 𝐴𝐵 .

If we connect A and B to a circle’s center O, we get an equilateral triangle, since we

know that AB equals the radius of the circle. Therefore the central angle AOB

measures 60 degrees, and so the arc 𝐴𝐵 represents one-sixth of the circle’s perimeter.

This of course holds for the second circle as well, as it has the same radius.

So the anwer is 5

6 of the two perimeters, or

5

6× 2 × 2 ∙ 𝜋 ∙ 𝑅 =

10

3∙ 𝜋 ∙ 10 =

100

3∙ 3.141592 = 104.7197… ≅ 104.72 𝑖𝑛𝑐𝑕𝑒𝑠.

Answers

1. 107,461

2. 100

3. 104.72

O

B

A

EJ

F

D

A

I

H

G

B

C

Category 2 You may use a calculator today!

Geometry Meet #4, February 2008

1. In the semi-circle to the right, Point O is the

center, and Points A and J are at opposite ends of

a diameter. The 8 points B, C, D, E, F, G, H, and

I are equally spaced around the semicircle and each is connected to both ends of

the diameter AJ. What is the sum of the degrees in the measures of the angles

�ABJ, �ACJ, �ADJ, �AEJ, �AFJ, �AGJ, �AHJ, and �AIJ?

2. In the square AIDF at the right two semicircles are drawn using AI and DF as diameters and the two semicircles are tangent to each other. AF = 8 cm. An ant crawls along the lines and arcs in a path that takes it from Point A and then through B, C, D, E, F, G, H, I, J, and back to A, in that order. How many centimeters long was the path the ant travelled? Express your answer as a decimal to the nearest tenth of a centimeter.

3. A circle is inscribed in the square to the left and the area between the two shapes is shaded. Using 3.14 as an estimation for . Jimmy calculated that the area of the shaded region is 104.06 cm2. Using 3.14 as an estimation for . again, how many centimeters are in the circumference of the circle? Express your answer as a decimal to the hundredths place.

Answers

1. _______________

2. _______________

3. _______________

I

E F

G

HC

D

B

O

A J

Solutions to Category 2

Geometry Meet #4, February 2008

1. All 8 of the triangles use the diameter as one side and have the 3rd vertex on the circle, so all 8 of the triangles are right triangles. The sum of the angles is just 8 � 90 � / 0

2. The path the ant travels will be along the entire circumference of the circle once and along the diameter twice. So it will travel a total of 8. � 8 � 8 1 2�. �

Here is what the path looks like with arrows to guide you.

3. If we call the radius of the circle r, then the sides of the square are all 2r making the area of the square (2r)2 = 4r

2. Since the area of the circle is 3.14r2, the

area of the shaded region must be 4r2 – 3.14r

2 = .86r2 which we know to be

104.06……So……..

. 864� � 104.06

4� � 121 4 � 11 and 5 � 22 Therefore the circumference of the circle is 3.14(22) = 69.08

Answers 1. 720 2. 41.1

3. 69.08

EJ

F

DI

A

H

G

B

C

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Category 2 Geometry Meet #4, February 2006 1. A toy car has wheels with a diameter of 1 inch. How many turns does each wheel make if the car rolls 12 feet across the floor? (There are 12 inches in 1 foot.) Use 3.14 for � and round your answer to the nearest whole number of turns. 2. How many centimeters are there in the circumference of a circle with an area of 36� square centimeters? Use 3.14 for � and express your answer as a decimal to the nearest tenth of a centimeter. 3. Three semi-circles of diameter 2 centimeters are cut from three sides of a 4-cm by 4-cm square to form the figure below. A circle of radius 1-cm is placed above the square without overlap. How many square centimeters are in the area of the figure? Use 3.14 for � and express your answer to the nearest tenth of a square centimeter.

Answers 1. _______________ 2. _______________ 3. _______________

You may use a calculator

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Solutions to Category 2 Geometry Meet #4, February 2006

1. Wheels with a diameter of 1 inch have a circumference of � × 1 = � inches, which is about 3.14 inches. If the car rolls 12 feet across the floor, then it rolls 12 × 12 = 144 inches. The question now is how many turns of 3.14 inches there are in 144 inches. Dividing 144 by 3.14, we get about 45.86 turns, which is 46 to the nearest whole number of turns. 2. The formula for the area of a circle is Acircle = πr 2 . We can find the radius of the given circle by solving the equation 36π = πr 2. Since 62 = 36, the radius must be 6 centimeters. The formula for the circumference of a circle is C = πD or 2πr , so the circumference of our circle is 12�. Using 3.14 as an approximation of �, we get 12 × 3.14 = 37.68 or 37.7 to the nearest tenth of a centimeter.

3. If we cut the circle above the square in half, we can fill two of the voids on the sides of the square That would leave a square with just one semicircular region cut out of it. The area of the square is 4 cm × 4 cm = 16 square centimeters. The area of a circle with radius 1 centimeter is � × 12 = � square centimeters, so the area of a semicircle with radius 1 cm is 0.5�. Thus the figure has an area of 16 – 0.5� = 16 – 1.57 = 14.43 or 14.4 square centimeters to the nearest tenth.

Answers 1. 46 2. 37.7 3. 14.4

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Category 2 Geometry Meet #4, February 2004 1. Segment AC is a diameter of the circle at right. If the measure of angle CAB is 49 degrees, how many degrees are in the measure of angle ACB? 2. A circle is inscribed in a square whose area is 2.25 square inches. How many square inches are there in the sum of the areas of the two shaded regions? Use 3.14 for � and express your answer as a decimal to the nearest hundredth. 3. Find the number of feet in the radius of a circle whose area given in square yards is numerically equivalent to its circumference given in feet. (Reminder: Three feet equals one yard.)

You may use a calculator

A

B

C

Answers 1. _______________ 2. _______________ 3. _______________

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Solutions to Category 2 Geometry Meet #4, February 2004

1. Since segment AC is a diameter, angle B must be a right angle. This means that angles A and C must add up to the other 90 degrees in the triangle. 90 – 49 = 41, so the measure of angle ACB must be 41 degrees. 2. If the area of the square is 2.25 square inches, its side length is the square root of 2.25, which is 1.5 inches.

2.25 = 1.5 ⇔ 1.52 = 2.25

The side length of the square and the diameter of the circle are the same. The formula for the area of a circle is A = πr 2, so we need the radius of the circle, which is 1.5 ÷ 2 = 0.75 inches. The area of the circle is thus Acircle = 3.14 × 0.752 = 3.14 × 0.5625 = 1.76625 square inches. Subtracting this from the area of the square, we get 2.25 – 1.76625 = 0.48375 square inches. The two shaded regions account for half of this difference, or 0.48375 ÷ 2 = 0.241875 square inches. Rounding this to the nearest hundredth, we get 0.24.

3. We have A = C, but A is in square yards and C is in square feet. Since we would like to write a single equation to solve for the unknown radius, we should convert the area of the circle from square yards to square feet. There are three feet in a yard, but nine square feet in one square yard (see picture at left). Our conversion will make the numerical value of the area nine times greater. We will have to multiply the value of the circumference by nine to keep these two quantities numerically equal. Thus we have the equation A = 9C . Since A = πr 2 and C = 2πr , we have πr 2 = 9 2πr( ). Dividing both sides by �r, we get r = 18 feet.

Answers 1. 41 2. 0.24 or .24 3. 18

One square yard equals

nine square feet.