Question

(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

Part a) minor arc: XY; major arc: XVY

Part b)
`248\°`

Part c) The tangent line is UV and the secant line is XY

Part d)
`UV=6√(5)\ units` or
`UV=13.42\ units`

Explanation:

Part a) we know that

The major arc is the larger arc joining two points on the circumference of a circle. Is an arc larger than a semicircle.

The sum of major and minor arcs is the whole circle, 360°

In this problem, possible major arcs are

VYX, VXY, XVY

and the corresponding minor arcs are

VX, VY, XY

Part b) we know that

The sum of major and minor arcs is the whole circle, 360°

so

Let

x------> measure minor arc

y-------> measure of major arc

`x+y=360\°`

we have

`x=112\°`

substitute

`112\°+y=360\°`

`y=360\°-112\°=248\°`

Part c) we know that

A tangent line intersects a circle at exactly one point.

In this problem

The tangent line is UV

A secant line intersects a circle in two points.

In this problem

The secant line is XY

Part d) we know that

The Intersecting Secant Theorem States: When two secant lines intersect each other outside a circle, the products of their segments are equal

so

`UV*UV = UX*UY`

`UV^(2)  = 9*(9+11)`

`UV^(2)  = 180`

`UV=√(180)\ units`

Simplify

`UV=6√(5)\ units` or
`UV=13.42\ units`