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Define a central angle in a circle.
An angle formed by two radii meeting at the circle's center. Its measure equals the measure of its intercepted arc.
Define an inscribed angle in a circle.
An angle formed by two chords that meet on the circle's circumference. Its measure is half the measure of its intercepted arc.
What is a chord?
A line segment connecting any two points on a circle's circumference.
Define a secant.
A line that intersects a circle at two points.
Define a tangent.
A line that touches a circle at exactly one point.
What is a cyclic quadrilateral?
A quadrilateral inscribed in a circle.
Define a circumscribed angle.
An angle formed by two tangent lines intersecting outside the circle.
State the Inscribed Angle Theorem.
Inscribed angles that intercept the same arc are congruent.
What is the relationship between a tangent and the radius at the point of tangency?
A tangent is always perpendicular to the radius drawn to the point of tangency, forming a right angle.
What is the measure of an angle inscribed in a semicircle?
An angle inscribed in a semicircle is always a right angle (90 degrees).
In a circle, arc AB measures 80 degrees. What is the measure of central angle AOB?
80 degrees, since the central angle equals the measure of its intercepted arc.
In a circle, arc CD measures 120 degrees. What is the measure of inscribed angle CED, where E is a point on the circle?
60 degrees, since the inscribed angle is half the measure of its intercepted arc.
Quadrilateral ABCD is inscribed in a circle. If angle A measures 85 degrees, what is the measure of angle C?
95 degrees, since opposite angles in a cyclic quadrilateral are supplementary (add up to 180 degrees).
Two tangent segments, PA and PB, are drawn to a circle from external point P. If PA = 7, what is the length of PB?
PB = 7, since tangent segments from the same external point are congruent.
A tangent line intersects a circle at point T. Radius OT is drawn. What is the measure of the angle between the tangent line and radius OT?
90 degrees, since a tangent is perpendicular to the radius at the point of tangency.
Secants PAB and PCD are drawn to a circle from external point P. PA = 3, AB = 5, and PC = 4. Find the length of CD.
Let CD = x. Using the Two Secants Theorem: (3+5)*3 = (4+x)*4. Solving for x, CD = 2.
Tangent PT and secant PAB are drawn to a circle from external point P. PT = 6 and PA = 3. Find the length of AB.
Let AB = x. Using the Secant-Tangent Theorem: 6^2 = (3+x)*3. Solving for x, AB = 9.
A circle has a radius of 8. A chord of length 12 is drawn. How far is the chord from the center of the circle?
Let the distance be d. A right triangle is formed with hypotenuse 8 and one leg 6 (half the chord length). Using the Pythagorean theorem: d^2 + 6^2 = 8^2. Solving for d, d = $\sqrt{28} = 2\sqrt{7}$
An angle is inscribed in a semicircle of radius 5. What is the area of the triangle formed by the angle's vertices?
Since the angle is inscribed in a semicircle, it's a right angle. The hypotenuse is the diameter (10), and the legs can vary. The maximum area occurs when the triangle is isosceles, with legs $5\sqrt{2}$. The area is (1/2) * base * height = (1/2) * $5\sqrt{2}$ * $5\sqrt{2}$ = 25.
Two circles intersect. Their common chord is extended to an external point P. A tangent from P to one circle has length 8. What is the length of the tangent from P to the other circle?
The tangents have the same length. Therefore, the length of the tangent from P to the other circle is 8.
How do you find the measure of an inscribed angle if you know the measure of its intercepted arc?
The measure of the inscribed angle is half the measure of its intercepted arc.
How is the measure of a central angle related to the measure of its intercepted arc?
The measure of a central angle is equal to the measure of its intercepted arc.
If two inscribed angles intercept the same arc, what can you conclude?
The two inscribed angles are congruent (equal).
If opposite angles of a quadrilateral add up to 180 degrees, what can you conclude?
The quadrilateral can be inscribed in a circle (it's a cyclic quadrilateral).
How do you find the measure of a circumscribed angle?
The measure of a circumscribed angle is half the difference of the measures of the intercepted arcs.
Describe the relationship between equal chords and their distance from the center of the circle.
Equal chords are equidistant from the center of the circle, and conversely, chords equidistant from the center are equal in length.
Explain the Two Secants Theorem.
For two secants drawn from an external point: $\text{(secant 1 length)} \times \text{(external segment 1)} = \text{(secant 2 length)} \times \text{(external segment 2)}$
Explain the Secant-Tangent Theorem.
For a secant and a tangent drawn from an external point: $\text{(tangent length)}^2 = \text{(secant length)} \times \text{(external segment)}$
How can you find the center of a circle if you know a chord?
The perpendicular bisector of any chord always passes through the circle's center.
How do you use the fact that a tangent is perpendicular to the radius at the point of tangency?
This creates a right triangle, allowing you to use the Pythagorean theorem or trigonometric ratios to solve for unknown lengths or angles.