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Welcome to the guide on Circles
Please note this topic is a big differentiator in getting a high score
There are no prerequisites for this lesson. Circles are such an important part of geometry that they deserve their own lesson. After following this tutorial, you will have the knowledge needed to solve circle problems for the TJ.
The first thing to know about circles that you may not have experienced is the idea of a radian. Radians are common measure of angles and, because of the way they are defined, often provide mathematical convenience in problems. Here’s an introduction to radians:
Radians and Degrees (An overview of the ways to express a degree measurement)
Here’s a terse overview of all the terms, definitions and formulas you will need to know:
Wiki (This site contains everything you must memorize about circles. Save this page to look at multiple times)
Guided Practice Problems:
Guided, adaptive learning (Khan Academy is a classic site to practice and hone in on your skills. Do as much as necessary to make you feel comfortable)
Detailed solutions (A visual instruction on circles to help you understand them from a different angle)
Additional problems (Overwhelming at first, this site is a very good summary of getting in the mindset of hunting for properties you know about circles)
Now check if you are battle-ready by answering the following questions:
1. The diagram shows a circle, center O and radius 5. AB is a chord of the circle of length 8. Calculate the distance from O to AB.
2. RS and RT are tangents to the circle center O. ∠SRT = 36°. What is the size of ∠SUT?
3. The diagram shows a circle, center O, and two chords AC and BD that intersect at a point P inside the circle. Arc AB subtends an angle of 144° at O and arc CD subtends an angle of 34° at O. What is the size of x°?
4. The diagram shows a circle, center O, radius 10 units. Chords AB and CD are parallel. The length of arc AB is 30 units and the length of arc CD is 20 units. What is the size of the angle, measured in radians, that arc AC subtends at O?
5. The diagram shows a circle with center O and radius 8 m. A, B and C are points on the circumference such that AB = AC and arc BC makes an angle of 150° at O. Calculate the area shaded blue. Give your answer to the nearest m2.
6. The diagram shows a circle with center O and radius 10 in. A and B are points on the circumference such that arc AB makes an angle of 135° at O. Calculate the area of the shaded segment. Give your answer to the nearest 0.1 inches squared. (note: it may be useful to use a calculator for this specific question)
7. A, B, C and D are points on the circumference of a circle, center O. Chords AB and CD intersect at the point X. ∠AXD = 92° and ∠CBA = 57°. What is the size of ∠DAX?
8. AB is a diameter of a circle, center O. C is a point on the circumference of the circle, such that ∠CAB = 2 × ∠CBA. What is the size of ∠CBA?
9. Points A, B, Q, D, and C lie on the circle shown and the measures of arcs BQ and QD are 42˚ and 38˚ respectively. The sum of the measures of angles P and Q is,
10. In a circle with center O, AD is a diameter, ABC is a chord, BO = 5, and ∠ABO = arc CD = 60˚. Then the length of BC is,
To look at the answers, click here