Secants and Tangents of a Circle: Lengths and Angles, Study Guides, Projects, Research of Physical education

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ANGENT AND SECANT OF A

CIRCLE

Given: circle

with two

secants

Find x:

a) 10

b) 15

c) 20

d) 28

Given: circle

with two

secants

Find x:

a) 17.

b) 30

c) 35.

d) 40

Given: circle with

tangent and

secant

Find x:

a) 1

b) 4

c) 8

d) 9

OPENING ACTIVITY

OPENING ACTIVITY

CIRCLES: SECANTS AND TANGENTS

Circle is an important and most common shape studied in geometry. It is being

seen in almost every subject.

We will learn how to relate the lengths of these segments mathematically.

INTRODUCTION

INTRODUCTION

There are many concepts related to circle. Tangent and Secant are quite useful ones.

A circle is all points equidistant from one point called the center of the circle. Segments drawn

within, through, or tangent to a circle create angles which we will now define and measure.

Intersecting segments also create smaller segments.

IMPORTANT FACTS:

IMPORTANT FACTS:

A line passing through two points on a

circle is called a SECANT. It cuts

the circle at two places. We may define

secant as a line segment that passes

through the circle.

A line that touches the circle at exactly

one point is called TANGENT.

a tangent is a line segment that is drawn

outside a circle and essentially

touches the circle at one and only point.

DISCUSSION

DISCUSSION

We show circle O below in figure a. Points A, B, C, and D are on the circle. The Segments AP an

DP are secants

Because they intersect the circle in two points. Notice that the arcs intercepted are arcs CB

and AD

How does the measure of angle P relate to

the arcs CB and AD?

By drawing the segments DC and AB shown

in red, we form the triangles ABP and DCP.

These are similar triangles

because they have angle P in common and

angles A and D must be equal because they

are inscribed angles

intercepting the same arc, CB. This means

that angles A and D must equal one half

the measure of arc CB. In figure a, we also show angle 1 which is

angle ACD because we will need to refer to

it below. Notice that angle 1 is inscribed

and intercepts arc AD.

Therefore, angle 1 has a measure equal to one half of ar

ANGLE outside formed by two secants:

ow in figure b, we only show triangle DCP from the circle diagram shown in figure a

As shown in the diagram, DCP is supplementary to angle 1.

The three angles of triangle DCP must have a

sum of 180 degrees.

Solving this equation for angle yeilds

This means that the measure of angle P, an angle external to a circle

and formed by two secants, is equal to one half the difference of the intercepted arcs.

MENTS formed by secants, drawn from a point, intersecting a

Figure a is shown again for reference.

We have already noted that triangles ABP

and CDP are similar, This gives

corresponding

sides as follows:

PC ~ PB and PD ~ PA

In a proportion true for

corresponding parts of similar

triangles, we have

hat these are the products of the exterior part of each secant with each secants entir

EGMENTS formed by a secant and a tangent, drawn from a po

intersecting a circle:

he case where one of the segments forming angle P is a tangent, we show the figure c again

We have added segments CB and

DC. Looking at triangles PCB and

PDC, we have the following:

i. Both triangles share angle P and

ii. Angle D and angle PCB both

have measure ½ arc CB, the

intercepted arc

Thus, triangles PCB and PDC are similar. Since

Sides PC ~ PD and sides PB

~ PC

We can write the proportion:

ii. In circle O below, secants are drawn from point P. PC = 10, PB = 9, AC = x, and DB =

What is the length of secant PA?

Notice that:

(PC)(PA) = (PB)(PD)

Which is the same as

10(10 + x) = 9(9 + 12) = 189

Which gives us

100 + 10x= 189

10x = 89

PA = 10 +x

PA = 10 +x

PA =

PA =

iii. In circle O below, two secants from point P intersect circle O such that arcs

CP = 10, BP = 9, CA = 2x, and BD = 2x + 3

What is the measure of segment AP?

The products of the external

segment and the entire secant must

be equal for both secants. We

have:

CP (CP + CA) = BP (BP + BD)

10(2x + 10) = 9(2x + 12)

Solving this equation for x we get:

20x + 100 = 18x +

2x = 8

x = 4

Since AP equals 2x + 10

AP = 2(4) + 10

AP = 2(4) + 10

AP = 18

AP = 18

v. The two secants in the picture below are not drawn to scale.

KO = 16, KJ = 4, and LO = 32

What is the measure of LM?

WE EVALUATE:

WE EVALUATE:

The angle formed outside a circle by intersecting secants ( or a

secant and tangent or two tangents ) is equal in measure to ½ the

difference of the intercepted arcs.

The angle formed outside a circle by intersecting secants ( or a

secant and tangent or two tangents ) is equal in measure to ½ the

difference of the intercepted arcs.

If two secants meet at a point outside a circle, the product of the

exterior part of one secant with its entire length is equal to the

product of the exterior part of the other secant with its entire

length.

If two secants meet at a point outside a circle, the product of the

exterior part of one secant with its entire length is equal to the

product of the exterior part of the other secant with its entire

length.

If a secant and a tangent meet at a point outside a circle, the

product of the exterior part of the secant with its entire length is

equal to the square of the tangent segment.

If a secant and a tangent meet at a point outside a circle, the

product of the exterior part of the secant with its entire length is

equal to the square of the tangent segment.