Question

. Given the following diagram. (a) Name the minor arc __________ Name the major arc ___________ (b) If the minor arc is 112°, what is the measure of the rest of the circle arc length? _____________ (c) Which line is the tangent? ___________ Which line is the secant? __________ (d) If XY is 11 and UX is 9, what is the length of UV? Show all work or no credit.

Answer 1

(a) minor arc: arc VX

major arc: arc VYX

(b) 248 degrees

(c) Tangent: UV

Secant: UY

Explanation:

(b) 360 - 112 = 248

(c) UV crosses on the circumference of the circle at exactly one point

UY crosses through the circle at exactly two points

Answer 2

• (a) minor arc: XY; major arc: XVY
• (b) XVY = 248°
• (c) tangent: UV; secant: UX
• (d) UV = 6√5

Explanation:

(a) Any pair of points on the circle that are separated by less than the diameter will define a minor arc. (The minor arc is the shortest arc of the circle between the points.) Possible minor arcs in this diagram are ...

VX, VY, XY

The corresponding major arc is usually named by adding the name of a point between the two endpoints that is not on the minor arc. For the minor arcs above, the corresponding major arcs are ...

VYX, VXY, XVY

Given that part (b) tells us the minor arc of interest is 112°, we assume that arc is the one subtended by the chord: XY.

Then, per the discussion above, the corresponding major arc is XVY.

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(b) The sum of major and minor arcs is the whole circle, 360°. So, the measure of the major arc is ...

360° -112° = 248°

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(c) A tangent line intersects a circle at exactly one point. It is perpendicular to a radius to that point of intersection. The tangent line in this diagram is UV.

A secant intersects a circle in two places. The portion of the secant between the points of intersection is called a chord. The secant line UX contains the chord XY.

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(d) A rule of secants (and chords) is that the product of distances from where the secants (or chords) meet to the two intersection points with the circle is the same. For a tangent line, effectively, the two points of intersection are at the same distance. This means ...

UV·UV = UX·UY

UV² = 9·(9+11) = 180

UV = √180 = √(6²·5)

UV = 6√5 ≈ 13.42

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The attached figure is drawn to scale with arc XY being 112°.