Question

GIVEN:

PB tangent
PV, PU secants

If m VU= 80° and m ST= 40°, then ∠1 =
If m VU= 70° and m ST= 30°, then ∠2 =
If m VB= 60° and m BS = 30°, then ∠3 =
If VS = 9, SP = 12 and UT = 4, then TP =

Answer 1

1= 40

2=50

3=15

TP= 14

Answer 2

Part a) If m VU= 80° and m ST= 40°, then ∠1 =

we know that

The measurement of the external angle is the semi-difference of the arcs it comprises.

so

`Angle (1)=(1)/(2)*(80\°-40\°)=20\°`

therefore

the answer Part a) is

∠1 =
`20\°`

Part b) If m VU= 70° and m ST= 30°, then ∠2 =

we know that

The measure of the internal angle is the semi- sum of the arcs comprising it and its opposite.

so

`Angle (2)=(1)/(2)*(70\°+30\°)=50\°`

the answer Part b) is

∠2 =
`50\°`

Part c) If m VB= 60° and m BS = 30°, then ∠3 =

we know that

The measurement of the external angle is the semi-difference of the arcs it comprises.

so

`Angle (3)=(1)/(2)*(60\°-30\°)=15\°`

therefore

the answer Part c) is

∠3 =
`15\°`

Part d) If VS = 9, SP = 12 and UT = 4, then TP =

we know that

The Intersecting Secants Theorem states that when two secant lines intersect each other outside a circle, the products of their segments are equal.

so

`PT*PU=PS*PV`

we have

`PS=12\ units\\PV=PS+VS=12+9=21\ units\\PU=PT+UT=PT+4`

substitute

`PT*(PT+4)=12*21`

`PT^(2) +4PT-252=0`

using a graphing tool------> to resolve the second order equation

see the attached figure

the solution is

`PT=14\ units`

therefore

the answer Part d) is

`PT=14\ units`