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 =

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

therefore

the answer Part a) is

∠1 =

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

the answer Part b) is

∠2 =

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

therefore

the answer Part c) is

∠3 =

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

we have

substitute

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

see the attached figure

the solution is

therefore

the answer Part d) is

1= 40

2=50

3=15

TP= 14

step-by-step explanation:

we have

using a graphing tool

see the attached figure

this is a quadratic equation (vertical parabola) open up

the vertex is a > vertex

the domain of the function is all real > interval (-∞,∞)

the range of the function is the > [-6.125,∞)

the function has two x->   and

the function is increasing over the > (0.25,∞)

the function is decreasing over the > (-∞,0.25)