(a) Using a ruler and a pair of compasses only, construct triangle ABC with /AB/ = 7.5cm, /BC/ = 8.1cm and < ABC = 105°. (b) Locate a point D on BC such tha...
Question 1 Report
(a) Using a ruler and a pair of compasses only, construct triangle ABC with /AB/ = 7.5cm, /BC/ = 8.1cm and < ABC = 105°.
(b) Locate a point D on BC such that /BD/ : /DC/ is 3 : 2.
(c) Through D, construct a line I perpendicular to BC.
(d) If the line I meets AC at P, measure /BP/.
Answer Details
Construction and measurement
- Draw \(BC=8.1\text{ cm}\).
- At \(B\), construct an angle of \(105^\circ\). Construct \(90^\circ\) and \(120^\circ\), then bisect the angle between them to obtain \(105^\circ\).
- On the \(105^\circ\) ray, use compasses to mark \(BA=7.5\text{ cm}\). Join \(A\) to \(C\).
- Divide \(BC\) in the ratio \(3:2\). Since there are \(3+2=5\) equal parts, \[ BD=\frac{3}{5}\times 8.1=4.86\text{ cm}. \] Mark \(D\) \(4.86\text{ cm}\) from \(B\) along \(BC\).
- Construct the perpendicular to \(BC\) through \(D\). Where it meets \(AC\) is \(P\).
- Measure \(BP\).
The measurement from \(B\) to \(P\) is approximately
\[ \boxed{BP\approx 5.4\text{ cm}} \]Examination reminder: The ratio \(3:2\) means that \(BC\) is split into \(5\) equal parts altogether, with \(D\) located \(3\) of those parts from \(B\), not \(2\) parts from \(B\).
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