Geometry: Circles
Angles, arcs, chords, tangents, and circle equations
Geometry: Circles
Angles, arcs, chords, tangents, and circle equations
Geometry - Grade 9-12
- 1
A circle has radius 7 cm. Find its circumference and area. Leave both answers in terms of pi.
The circumference is 14π cm because C = 2πr = 2π(7). The area is 49π square cm because A = πr² = π(7²). - 2
In circle O, central angle AOB measures 128 degrees. What is the measure of minor arc AB?
A central angle has its vertex at the center of the circle.
The measure of minor arc AB is 128 degrees because the measure of a minor arc equals the measure of its central angle. - 3
An inscribed angle intercepts an arc that measures 86 degrees. Find the measure of the inscribed angle.
Use the theorem: inscribed angle = one half of intercepted arc.
The inscribed angle measures 43 degrees because an inscribed angle is half the measure of its intercepted arc. - 4
Points A, B, C, and D lie on a circle. If angle ABC measures 52 degrees, what is the measure of arc AC that does not contain point B?
The arc AC that does not contain point B measures 104 degrees because an inscribed angle is half the measure of the arc it intercepts. - 5
A tangent line touches a circle at point T. Radius OT is drawn to the point of tangency. If angle OTP is formed by the radius and the tangent line, find its measure.
A tangent line meets a circle at exactly one point.
Angle OTP measures 90 degrees because a radius drawn to the point of tangency is perpendicular to the tangent line. - 6
From the same external point P, two tangent segments are drawn to a circle. The tangent segments are PA and PB. If PA = 3x + 4 and PB = 5x - 10, find x and the length of each tangent segment.
Set the two tangent segment expressions equal to each other.
The value of x is 7 because tangent segments from the same external point are congruent, so 3x + 4 = 5x - 10. Each tangent segment is 25 units long because 3(7) + 4 = 25 and 5(7) - 10 = 25. - 7
A chord is 16 inches long and is 6 inches from the center of a circle. Find the radius of the circle.
Draw a right triangle using half the chord, the distance from the center, and the radius.
The radius is 10 inches. The perpendicular from the center to a chord bisects the chord, so half the chord is 8 inches. Using the right triangle, r² = 6² + 8² = 36 + 64 = 100, so r = 10. - 8
Two chords intersect inside a circle. One chord is split into segments of lengths 4 and 15. The other chord is split into segments of lengths 6 and x. Find x.
For intersecting chords, multiply the two parts of one chord and set that equal to the product of the two parts of the other chord.
The value of x is 10. By the intersecting chords theorem, 4 · 15 = 6 · x, so 60 = 6x and x = 10. - 9
Two secants are drawn from the same external point. The first secant has an external segment of 5 and a whole length of 20. The second secant has an external segment of 8 and a whole length of x. Find x.
Use external segment times whole secant length for each secant.
The value of x is 12.5. By the secant-secant theorem, 5 · 20 = 8 · x, so 100 = 8x and x = 12.5. - 10
A tangent and a secant are drawn from the same external point. The tangent segment is 12 units long. The secant has an external segment of 8 units and a whole length of x units. Find x.
The whole secant length is 18 units. By the tangent-secant theorem, 12² = 8x, so 144 = 8x and x = 18. - 11
Find the length of a 60 degree arc in a circle with radius 9 cm. Leave your answer in terms of pi.
Arc length is the same fraction of the circumference as the central angle is of 360 degrees.
The arc length is 3π cm. Use s = (θ/360) · 2πr, so s = (60/360) · 2π(9) = (1/6) · 18π = 3π. - 12
Find the area of a sector with central angle 120 degrees in a circle with radius 6 meters. Leave your answer in terms of pi.
The sector area is 12π square meters. Use A = (θ/360) · πr², so A = (120/360) · π(6²) = (1/3) · 36π = 12π. - 13
Write the equation of a circle with center (3, -2) and radius 5.
Be careful with the signs of h and k in the standard form.
The equation is (x - 3)² + (y + 2)² = 25. A circle with center (h, k) and radius r has equation (x - h)² + (y - k)² = r². - 14
Find the center and radius of the circle given by the equation (x + 4)² + (y - 1)² = 49.
The center is (-4, 1), and the radius is 7. In standard form, (x - h)² + (y - k)² = r², so h = -4, k = 1, and r = √49 = 7. - 15
Complete the square to write x² + y² - 6x + 8y - 11 = 0 in standard form. Then identify the center and radius.
Move the constant to the other side first, then add the square of half each linear coefficient to both sides.
The standard form is (x - 3)² + (y + 4)² = 36. The center is (3, -4), and the radius is 6. This comes from x² - 6x = (x - 3)² - 9 and y² + 8y = (y + 4)² - 16.