TANGENTS

In the figure above, ↔ and ↔ are two tangents intersecting outside the circle at point F. Arc GCJ and arc GJ are the two

FB FEintercepted arcs of L GFJ.

a. If arc GCJ measures 200° and arc GJ measures 160°, what is the measure of L GFJ?

mL GFJ = ½ (arc GCJ – arc GJ)mL GFJ = ½ (200° - 160°)mL GFJ = ½ (40°)mL GFJ = 20 °

b. ↔ and ↔ are tangents. What is the measure of ↔ if ↔ measures 20 inches? FB FE FE FB

FE = 20 inches; the measure of ↔ is the same with the measure of ↔ because they are congruent. FE FB

c. ↔ and ↔ are tangents. The measure of ↔ is 35 cm and the measure of ↔ is x^2 + 10. FB FE FB FEFind x.

x^2 + 10 = 35 -10 -10√x^2 = √25 x = 5

d. ↔ and ↔ are tangents. The measure of arc GCJ measures 162° and the measure of L GFJ is 72°, what is the measure of the

FB FEother arc which is arc BE?

72° = ½ (162° - arc BE)2 · 72° = 2 · ½ (162° - arc BE) 144° = 162° – arc BEarc BE = 162° – 144°arc BE = 18 °

e. Suppose the measure of ↔ is 6x + 5 and the measure of ↔ is 4x + 15. Find: FB FE

I. xII. ↔ FBIII. ↔ FE

I. 6x + 5 = 4x + 15 - 4x -4x 2x + 5 = 15 2x = 20 x = 10

II. 6x + 5 = ↔ FB

6(10) + 5 = ↔ FB

60 + 5 = ↔ FB

65 = ↔ FB

III. 4x + 15 = ↔

FE 4(10) + 15 = ↔

FE 40 + 15 = ↔

FE 55 = ↔

FE

SECANTS

In the figure above, ↔ and ↔ are two secants intersecting outside the circle at point F. arc CD and arc HI are the two

CH DI

intercepted arcs of L CFD.

a. If arc CD measures 130° and arc HI measures 60°, what is the measure of L CFD?

mL CFD = ½ (arc CD – arc HI)mL CFD = ½ (130° - 60°)mL CFD = ½ (70°)mL CFD = 35 °

b. In circle A above, two secants from point F intersect circle A such that arcs FH = 8, FI = 10, HC = 3x, and ID = 2x +5. What is the measure of segment CF?

The products of the external segment and the entire secant must be equal for both secants. We have:

FH (FH + HC) = FI (FI + ID)8 (3x + 8) = 10 (2x + 15)

Solving this equation for x we get

24x + 64 = 20x + 1504x = 86x = 21.5

Since CF equals 3x + 8

CF = 3(21.5) + 8CF = 72.5

c. In circle A above, secants FC and FD are drawn from point F. We have the following measurements given: FH = 6, HC = 8, FI = x + 2, and ID = 5x +7. What is the measure of chord ID?

FH (FC) = FI (FD)6 (14) = (x + 2)(6x + 9)

Solving this equation for x gives us

6(14) = (x + 2)(6x + 9)84 = 6x^2 + 21x + 180 = 6x^2 + 21x - 660 = 3(2x^2 + 7x - 22)0 = 3(x - 2)(2x + 11)x = 2 and x = -11/2

Using x = 2, we have

ID = 5(2) + 7 = 17

Using x = -11/2, we have

ID = 5(-11/2) + 7 = -41/2

We must discard this answer since the length of a segment cannot be a negative number.

d. In circle A shown above, two secants from point F intercept arcs HI = x – 20 and CD = 2x. What is the measure of arc

CD if angle F is 75°?

We know that the measure of an external angle F when formed by two secants is equal to one half the difference of the

measures of the intercepted arcs.

mL F = ½ (arc CD – arc HI)mL F = ½ [2x – ( x – 20) ]mL F = ½ (x + 20)75 = (1/2)(x + 20)150 = x + 20x = 130º

Since we were given that arc CD = 2x

CD = 2 (130º)CD = 260°

e. In circle A above, angle F is x – 5, arc HI is 40, and arc CD is 4x. What is the measure of angle F?

CD and HI are arcs intercepted by the secants FC and FD. The measure of angle F must be ½ the difference of the measures of

arcs CD and HI.

mL F = ½ (arc CD – arc HI)x – 5 = ½ (4x – 40)

Solving for x:

2x –10 = 4x – 404x – 2x = 40 – 10

2x = 40 – 102x = 30x = 15

We were given that angle F = x – 5, substituting x = 15

mL F = 15 – 5mL F = 10 °

Activity 10: My True World!Make a design of an arch bridge that would connect two places which are

separated by a river, 20 m wide. Indicate

on the design the different measurements

of the parts of the bridge. Out of the design and the measurements of its parts,

formulate problems involving tangent and secant segments, and then solve. Use the rubric provided to rate your work.

Sophia Marie D. Verdeflor Grade 10-1 STE

Activity 10:

River: 20 m wide

8 m

10 m