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1 Name: ___________________________________
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Name: ___________________________________
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Section 8.1 – Properties of Tangents to a Circle
A tangent is a line that intersects a circle at only one point. This point is called the point of tangency.
In the diagram, line AB is a tangent to the circle with center O. Point P is the point of tangency.
Tangent-Radius Property
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When using this property, you should always be looking to find a right triangle for two important reasons:
1. The angles in a triangle add to equal 180° 2. The lengths of the sides of a right triangle are related using the Pythagorean Theorem
Recall: 𝒉𝟐 = 𝒍𝟐 + 𝒍𝟐 𝒐𝒓 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
1. In each diagram, point O is the centre of the circle, which lines are tangents?
2. Point P is the point of tangency in each of the following circles with center O. Determine the missing measure in each.
a)
𝑥° = _____
b)
𝑥° = _____
c)
𝑥° = _____
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d)
𝑎 = _____
e)
𝑎 = _____
f)
𝑎 = _____
3. Determine the missing measures in each of the following diagrams. Point O is the center of each circle.
a)
𝑑° = _____
𝑒° = _____
b)
𝑎 = _____
𝑏 = _____
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4. A small aircraft, A, is cruising at an altitude of 1.5 km. The radius of the Earth is approximately 6400 km. How far is the plane from the horizon at B? Calculate to the nearest kilometer.
5. Point O is the center of the circle. Point B is a point of tangency. Determine the missing measures. (Round to the nearest tenth where necessary).
𝑥 = _____
𝑦 = _____
𝑧° = _____
6. A circular mirror with radius 20 cm hangs by a wire from a hook. The wire is 30 cm long and is tangent to the mirror in two places. How far above the top of the mirror is the hook?
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Section 8.2 – Properties of Chords in a Circle
A chord is a line segment that joins two points on a circle. The diameter of a circle is a chord through the center of the circle.
A perpendicular bisector intersects a line segment at 90° and divides the line segment into two congruent (equal) parts.
The chord, its perpendicular bisector, and the center of the circle are related.
Perpendicular to Chord Properties
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When using these properties, you will likely have to add something to the diagram. You will be looking to construct a right triangle so you can again use the Pythagorean Theorem!
7. Determine the missing measures indicated in the diagrams below. Round to the nearest tenth when necessary.
a)
𝑒 = _____
b)
𝑓 = _____
c)
𝑥° = _____
𝑦° = _____
d)
𝑥° = _____
𝑦° = _____
e)
𝑥° = _____
𝑦° = _____
f)
𝑎 = _____
𝑏 = _____
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g)
𝑎 = _____
𝑏 = _____
h)
𝑏 = _____
i)
𝑟 = _____
j)
𝑟 = _____
k)
𝑠 = _____
l)
𝑠 = _____
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8. A circle has a diameter of 25 cm. How far from the center of the circle is a chord 16 cm long? Justify your answer.
9. A chord is 6 cm long and is 15 cm from the center of a circle. What is the radius of the circle? What is the diameter?
10. A pedestrian underpass is constructed beneath a roadway using cylindrical pipe with radius 1.8 m. The bottom of the pipe will be filled and paved. The headroom at the center of the path is 2.8 m. How wide is the path?
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Section 8.3 – Properties of Angles in a Circle
An arc is a section of the circumference of a circle:
x A minor arc is less than half of the length of the circumference x A major arc is more than half of the length of the circumference
A central angle is formed by joining the endpoints of an arc to the center of the circle.
An inscribed angle is formed by joining the endpoints of an arc to a point on the circle.
In the diagram above, the inscribed and central angles are subtended by the minor arc AB. This means that they start and end in at the same points!
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Angle Properties
11. State an inscribed angle and central angle subtended by the same arc in each circle. a) b)
Central Angle: ________ Central Angle: ________
Inscribed Angle: ________ Inscribed Angle: ________
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12. Determine the missing measures indicated in the diagrams below.
a)
𝑥° = _____
b)
𝑥° = _____
c)
𝑥° = _____
𝑦° = _____
d)
𝑥° = _____
𝑦° = _____
e)
𝑥° = _____
𝑦° = _____
f)
𝑦° = _____
𝑧° = _____
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g)
𝑦° = _____
𝑧° = _____
h)
𝑥° = _____
𝑦° = _____
i)
𝑥° = _____
𝑦° = _____
j)
𝑥° = _____
𝑦° = _____
k)
𝑥° = _____
𝑦° = _____
l)
𝑥° = _____
𝑦° = _____
