In the diagram, \(\overline{RT}\) and \(\overline{RT}\) are tangent to the circle with centre O. < TUS = 68 °, < SRT = x, and < UTO = y. Find the value of x...
Question 1 Report
In the diagram, \(\overline{RT}\) and \(\overline{RT}\) are tangent to the circle with centre O. < TUS = 68 °, < SRT = x, and < UTO = y. Find the value of x.
(b) Two tanks A and B am filled to capacity with diesel. Tank A holds 600 litres of diesel more than tank B. If 100 litres of diesel was pumped out of each tank, tank A would then contain 3 times as much diesel as tank B. Find the capacity of each tank.
Answer Details
(a) Finding x
From the diagram, \(RT\) and \(RS\) are tangents drawn from the external point R, touching the circle (centre O) at T and S. U lies on the circle, with the inscribed angle \(\angle TUS = 68^\circ\), and \(\angle SRT = x\).
The inscribed angle \(\angle TUS\) stands on the arc \(TS\) that faces R (the arc not containing U). By the inscribed-angle theorem, the central angle on that same arc is
\[\angle TOS = 2\times \angle TUS = 2\times 68^\circ = 136^\circ\]In quadrilateral \(OTRS\), the radii meet the tangents at right angles, so \(\angle OTR = \angle OSR = 90^\circ\). The angles of the quadrilateral sum to \(360^\circ\):
\[\angle TOS + \angle OTR + \angle SRT + \angle OSR = 360^\circ\]\[136^\circ + 90^\circ + x + 90^\circ = 360^\circ\]\[x = 360^\circ - 316^\circ = 44^\circ\]Therefore \(x = 44^\circ\).
(b) Capacity of the two diesel tanks
Let the capacity of tank B be \(b\) litres. Then tank A holds \(b + 600\) litres.
After pumping out 100 litres from each, tank A contains 3 times as much as tank B:
\[(b + 600) - 100 = 3\big(b - 100\big)\]\[b + 500 = 3b - 300\]\[500 + 300 = 3b - b\]\[800 = 2b \;\Rightarrow\; b = 400\]So tank B holds \(400\) litres and tank A holds \(400 + 600 = 1000\) litres.
Check: after pumping, A = \(900\), B = \(300\), and \(900 = 3\times 300\). Correct.
Tank A has a capacity of \(1000\) litres and tank B has a capacity of \(400\) litres.
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