Geometry Question

Question

In the diagram, lines AB and CB are tangent to the circle at points A and C, respectively. They intersect outside of the circle at point B to form angle CBA, which intercepts major arc APC and arc AC. If the angle measure of CBA=60, the major arc measure of APC=(9x-30), and arc measure of AC=(4x), what is the value of x?

24

36

21

30

27

Answer 1

Hello Jacob,

Suppose, as in your problem, that one line is tangent to a given circle at point A, and a second line is tangent to this circle at point C. Suppose further that the lines intersect at the point B (so we name the lines AB and CB.) Let θ be the measure of ∠CBA (in degrees.) We are given that θ = 60º. Let us also use degree measure to measure arcs. (That is, the measure of a given arc on a circle will be the same as the measure of the central angle that cuts off this arc.) Note that the points A and C cut off two arcs on the circle. The smaller is (called arc AC) has measure less than 180º. The larger arc (major arc) is called arc APC, and it has measure greater than 180º. The following key fact gives the relationship between the measure of ∠CBA and the measure of arc AC is

Measure(Arc AC) = 180º - θ

Given that θ = 60°, and Measure(Arc AC) = 4x, we have

4x = 180º - 60º

4x = 120º

x = 30º

Hope that helps! Let me know if you need any further explanation.

William

PS. Note that you can also find x using the information about the major arc. Since the measure of arc AC is 180º - θ, then Measure(Arc APC) = 360º- (180º- θ) = 180º + θ = 240º. Set this equal to 9x - 30 and solve for x, and you will obtain the same result as above.

Answer 2

Question: In the diagram, lines AB and CB are tangent to the circle at points A and C, respectively. They intersect outside of the circle at point B to form angle CBA, which intercepts major arc APC and arc AC. If the angle measure of CBA=60, the major arc measure of APC=(9x-30), and arc measure of AC=(4x), what is the value of x?

Answer: We will use circle theorems to solve for x.

Step 1: Identify the circle theorem.

Far arc - near arc divided by 2 equals the angle formed outside the circle by the intersection of two tangent lines, a tangent and a secant or two secants.

Step 2: Use the theorem to find x.

Far arc is arc APC. Near arc is arc AC. The angle (angle ABC) measures 60 degrees.

arc APC - Arc AC

———————— = angle ABC

2

(9x-30) - (4x)

—————— = 60

2

(9x-30) - (4x)

—————— * 2 = 60 * 2

2

9x-30-4x = 120

5x-30 = 120

+30 +30

––––––––––

5x = 150

–– –––

5 5

x = 30

We could have done it using another circle theorem.

Step 1: Identify the circle theorem.

A circle has 360 degrees. The sum of the measures of its arcs should equal 360.

Step 2: Use the theorem to solve for x.

The arcs that make up the circle's circumference are arc APC and arc AC.

arc APC + arc AC = 360

( 9x - 30) + (4x) = 360

9x - 30 +4x = 360

13x - 30 = 360

+30 +30

––––––––––––

13x = 390

––– ––––

13 13

x = 30