Mathematics 10 — Exercise 7.2 · Q2 (vi)

Full question

Calculate the area of triangle ABC using the following measurements.

Step-by-step solution

1. Angles of a triangle sum to 180°. Subtract both known angles.

   $$B=180-23.0-89.0$$

2. Calculate the remaining angle.

   $$B=68.0$$

3. Rearrange the sine law to isolate the missing side.

   $$a=\frac{6.6\times \sin \left(\mathrm{23.0°}\right)}{\sin \left(\mathrm{89.0°}\right)}$$

4. Evaluate both sines and multiply the known side by the numerator sine.

   $$a=\frac{2.5788254}{0.9998477}$$

5. Divide by the denominator sine.

   $$a=2.5792183$$

6. Rearrange the sine law to isolate the missing side.

   $$b=\frac{6.6\times \sin \left(\mathrm{68.0°}\right)}{\sin \left(\mathrm{89.0°}\right)}$$

7. Evaluate both sines and multiply the known side by the numerator sine.

   $$b=\frac{6.1194134}{0.9998477}$$

8. Divide by the denominator sine.

   $$b=6.1203456$$

9. Use half the product of two sides and the sine of their INCLUDED angle.

   $$\mathrm{area}=\frac{15.785707\times \sin \left(\mathrm{89.0°}\right)}{2}$$

10. Evaluate the sine first.

    $$\sin \left(\mathrm{89.0°}\right)=0.9998477$$

11. Multiply the two side lengths.

    $$2.5792183\times 6.1203456=15.785707$$

12. Multiply by the sine, then divide by 2.

    $$\frac{15.783303}{2}$$

13. The area is in square centimetres.

    $$\mathrm{area\; (cm²)}=7.8916515$$

Answer

The final answer is shown below.