Question

1. find the length of side ab using trigonometry. note: abc is not a right triangle. (hint: draw 2 right triangles within abc) (7 marks)

Explanation:

Draw altitude from C to AB

Let \(D\) be the foot of the altitude drawn from vertex \(C\) to side \(AB\). This splits \(\triangle ABC\) into two right-angled triangles: \(\triangle ADC\) and \(\triangle BDC\), with right angles at \(D\).

Calculate segments in triangle ADC

In right-angled \(\triangle ADC\), the hypotenuse is \(AC = 16.92\text{ m}\) and \(\angle CAD = 46^\circ\).
Using basic trigonometric ratios:

$$ AD = AC \cdot \cos(46^\circ) = 16.92 \cdot \cos(46^\circ) $$

$$ CD = AC \cdot \sin(46^\circ) = 16.92 \cdot \sin(46^\circ) $$

Calculating these values:

$$ AD \approx 16.92 \cdot 0.694658 \approx 11.75\text{ m} $$

$$ CD \approx 16.92 \cdot 0.719340 \approx 12.17\text{ m} $$

Calculate segment BD in triangle BDC

In right-angled \(\triangle BDC\), the hypotenuse is \(BC = 27.72\text{ m}\) and the altitude is \(CD \approx 12.17\text{ m}\).
Using the Pythagorean theorem:

$$ BD = \sqrt{BC^2 - CD^2} $$

$$ BD = \sqrt{27.72^2 - 12.17^2} $$

$$ BD \approx \sqrt{768.40 - 148.11} = \sqrt{620.29} \approx 24.91\text{ m} $$

Find total length of AB

The total length of side \(AB\) is the sum of the segments \(AD\) and \(BD\):

$$ AB = AD + BD $$

$$ AB \approx 11.75 + 24.91 = 36.66\text{ m} $$

Answer:

\(36.66\text{ m}\)