Question

Please answer
Picture attached

Answer 1

1)a)62

1)b)31

Explanation:

1)a)

∠MTS = 31

Also, 2* ∠MTS = ∠MOS

∠MOS = 2*31 = 62

b) MS = MP

so ∠MTS = ∠MLP

∠MLP = 31

2)NL = PL/2

OL = ML/2

in triangle LMP and LON,

∠MLP = ∠OLN

by SAS, the triangles are similar

ML/OL = MP/ON

ML/(ML/2) = MP/ON

MP/ON = 2

ON = MP/2

Answer 2

1) a) ∠MOS = 62

b) ∠MLP = 31

Explanation:

1)

a) Chord MS creates an angle of ∠MTS = 31 at a point on the circle

∠MOS is the angle created by chord MS at the centre

Therefore,

∠MOS = 2*∠MTS

= 2*31

=62

∠MOS = 62

b) Since PM = MS, the angle created by these chords are the same

∠MLP = ∠MTS

∠MLP = 31

2) Prove that ON = MS/2

Since MS = MP,

we need to prove that ON = MP/2

We have OL = OM = radius

OL + OM = ML

ML = 2*OL -eq(1)

Since ON bisects LP,

LN = NP

LP = LN + NP

LP = 2*LN -eq(2)

Consider ΔMLP and ΔOLN

`(ML)/(OL) = (2*OL)/(OL) =2\;\;(from\;eq(1))\\\\(LP)/(LN) = (2*LN)/(LN) =2\;\;(from\;eq(2))\\\\\implies (ML)/(OL) = (LP)/(LN)`

Also ∠MLP = ∠OLN = 31

By SAS property, ΔMLP and ΔOLN are similar

`\implies (ML)/(OL) = (LP)/(LN) = (MP)/(ON)\\\\\implies (MP)/(ON) = 2\\\\\implies (MP)/(2) = ON`