Mathematics 10 — Exercise 7.2 · Q2 (viii)

Full question

Calculate the area of triangle ABC using the following measurements.

Step-by-step solution

1. Use cosine law to find this angle’s cosine.

   $$\cos \left(\mathrm{A°}\right)=\frac{{8.34}^2+{7.12}^2-37.4544}{2\times 8.34\times 7.12}$$

2. Calculate the squared terms, subtract, then divide by the denominator.

   $$\cos \left(\mathrm{A°}\right)=0.697158$$

3. Take inverse cosine in DEGREE mode.

   $$A=45.800567$$

4. Use cosine law to find this angle’s cosine.

   $$\cos \left(\mathrm{B°}\right)=\frac{{6.12}^2+{7.12}^2-69.5556}{2\times 6.12\times 7.12}$$

5. Calculate the squared terms, subtract, then divide by the denominator.

   $$\cos \left(\mathrm{B°}\right)=0.21335004$$

6. Take inverse cosine in DEGREE mode.

   $$B=77.681254$$

7. Subtract A and B from 180° to find C.

   $$C=56.518178$$

8. For three given sides use Heron’s formula. First find the semiperimeter.

   $$s=\frac{6.12+8.34+7.12}{2}$$

9. Add the three sides, then divide by 2.

   $$s=10.79$$

10. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 153}{1\times 25}$$

11. Calculate these two products separately.

    $$-\frac{153}{25}$$

12. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{153}{25}=-\frac{153}{25}$$

13. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=100$$

14. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1079}{100}=\frac{1079\times 1}{100\times 1}$$

15. Calculate the converted numerator; the denominator is now common.

    $$\frac{1079}{100}$$

16. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{153}{25}=\frac{\left(-153\right)\times 4}{25\times 4}$$

17. Calculate the converted numerator; the denominator is now common.

    $$-\frac{612}{100}$$

18. Keep the common denominator and add the signed numerators.

    $$\frac{1079-612}{100}=\frac{467}{100}$$

19. Calculate this operation before combining the remaining terms.

    $$\frac{1079}{100}-\frac{153}{25}=\frac{467}{100}$$

20. Subtract this side from s to obtain one Heron factor.

    $$10.79-6.12=4.67$$

21. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 417}{1\times 50}$$

22. Calculate these two products separately.

    $$-\frac{417}{50}$$

23. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{417}{50}=-\frac{417}{50}$$

24. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{417}{50}=\frac{\left(-417\right)\times 2}{50\times 2}$$

25. Calculate the converted numerator; the denominator is now common.

    $$-\frac{834}{100}$$

26. Keep the common denominator and add the signed numerators.

    $$\frac{1079-834}{100}=\frac{245}{100}$$

27. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

28. Show the two divisions explicitly.

    $$\frac{\frac{245}{5}}{\frac{100}{5}}$$

29. Evaluate the numerator division.

    $$\frac{245}{5}=49$$

30. Evaluate the denominator division.

    $$\frac{100}{5}=20$$

31. Write the reduced fraction.

    $$\frac{49}{20}$$

32. Calculate this operation before combining the remaining terms.

    $$\frac{1079}{100}-\frac{417}{50}=\frac{49}{20}$$

33. Subtract this side from s to obtain one Heron factor.

    $$10.79-8.34=2.45$$

34. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 178}{1\times 25}$$

35. Calculate these two products separately.

    $$-\frac{178}{25}$$

36. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{178}{25}=-\frac{178}{25}$$

37. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{178}{25}=\frac{\left(-178\right)\times 4}{25\times 4}$$

38. Calculate the converted numerator; the denominator is now common.

    $$-\frac{712}{100}$$

39. Keep the common denominator and add the signed numerators.

    $$\frac{1079-712}{100}=\frac{367}{100}$$

40. Calculate this operation before combining the remaining terms.

    $$\frac{1079}{100}-\frac{178}{25}=\frac{367}{100}$$

41. Subtract this side from s to obtain one Heron factor.

    $$10.79-7.12=3.67$$

42. Multiply all four factors, then take their positive square root.

    $$\mathrm{area}=\sqrt{10.79\times 4.67\times 2.45\times 3.67}$$

43. The area is in square centimetres.

    $$\mathrm{area\; (cm²)}=21.285568$$

Answer

The final answer is shown below.