Answer:
a. i) ∠TUV = 39°
ii) The measure of arc VTS (the minor arc) = 102°
b. The diameter <em>OT</em> of the circle = 12
c. The angle which we can calculate, given ∠MOP, is angle ∠OTM = 51°
i) Arc MVT = 98° and arc MO = 82°
Step-by-step explanation:
The given parameters are;
The radius of the circle = 6
∠SUT = 39°, ∠MOP = 49°
a. i) According to two tangent theory, two tangents that meet at a given point are congruent
Therefore, VU is congruent to SU
Given that PU is congruent to PU by reflexive property and PV = PS = The radius of the circle, we have;
ΔPVU is congruent to ΔPSU by Side Side Side (SSS) rule of congruency
∠SUT ≅ ∠TUV by Congruent Parts of Congruent Triangles are Congruent (CPCTC)
Therefore, ∠SUT = ∠TUV = 39° by transitive property of equality
ii) Arc VTS is the minor arc while arc VOS is the major arc by size
The arc measure that describes arc VTS is the minor arc
iii) From circle theorem, we have that the sum of the angle formed by two tangents and the minor arc equals 180°
Therefore, ∠SUT + ∠TUV + arc VTS = 180°
∴ Arc VTS = 180° - (39° + 39°) = 102°
b. The line segment length that can be calculated based on knowing the radius length includes the length of the diameter <em>OT</em> of the circle
The diameter <em>OT</em> = 2 × The length of the radius
∴ The diameter <em>OT</em> = 2 × 6 = 12
c. The angle ∠MOP, is an interior angle of the right triangle ΔTMO formed by the diameter of the circle, <em>OT</em>, therefore, given that ∠MOP = 39°, we have;
∠OTM = 90° - ∠MOP
∴ ∠OTM = 90° - 39° = 51°
∠OTM = 51°
Therefore, given ∠MOP, we can calculate angle ∠OTM
i) The arc that we can calculate, given ∠MOP are arc MVT and arc MO
Arc MVT = 2 × ∠MOP
∴ Arc MVT = 2 × 49° = 98°
Arc MO = 2 × ∠OTM
∴ Arc MO = 2 × (90° - 49°) = 82°
