Question

find the area of $\triangle abc$ to the nearest square unit.

1. $a = 71\\,\text{cm}, b = 23\\,\text{cm}, c = 84^\circ$
2. $b = 36\\,\text{ft}, c = 108^\circ, b = 29^\circ$
3. $a = 39\\,\text{m}, c = 63\\,\text{m}, b = 104^\circ$
4. $b = 376\\,\text{km}, c = 538\\,\text{km}, a = 73^\circ$

Explanation:

Problem 16: $b=28\ \text{m}, c=42\ \text{m}, A=126^\circ$

Step1: Use SAS area formula

The formula for area with two sides and included angle is $Area=\frac{1}{2}bc\sin A$.

Step2: Substitute given values

$$Area=\frac{1}{2} \times 28 \times 42 \times \sin(126^\circ)$$
Calculate $\sin(126^\circ)\approx0.8090$, then:
$$Area=\frac{1}{2} \times 28 \times 42 \times 0.8090 = 14 \times 42 \times 0.8090 = 588 \times 0.8090 \approx 476$$

---

Problem 17: $a=71\ \text{cm}, b=23\ \text{cm}, C=84^\circ$

Step1: Use SAS area formula

The formula for area with two sides and included angle is $Area=\frac{1}{2}ab\sin C$.

Step2: Substitute given values

$$Area=\frac{1}{2} \times 71 \times 23 \times \sin(84^\circ)$$
Calculate $\sin(84^\circ)\approx0.9945$, then:
$$Area=\frac{1}{2} \times 71 \times 23 \times 0.9945 = \frac{1}{2} \times 1633 \times 0.9945 \approx 816.5 \times 0.9945 \approx 812$$

---

Problem 18: $b=36\ \text{ft}, C=108^\circ, B=29^\circ$

Step1: Find angle $A$

Sum of angles in triangle: $A=180^\circ - B - C$
$$A=180^\circ - 29^\circ - 108^\circ = 43^\circ$$

Step2: Use Law of Sines to find side $c$

$$\frac{c}{\sin C}=\frac{b}{\sin B} \implies c=\frac{b\sin C}{\sin B}$$
$$c=\frac{36 \times \sin(108^\circ)}{\sin(29^\circ)} \approx \frac{36 \times 0.9511}{0.4848} \approx 70.9$$

Step3: Use SAS area formula

Use sides $b,c$ and included angle $A$: $Area=\frac{1}{2}bc\sin A$
$$Area=\frac{1}{2} \times 36 \times 70.9 \times \sin(43^\circ) \approx 18 \times 70.9 \times 0.6820 \approx 18 \times 48.35 \approx 870$$

---

Problem 19: $a=39\ \text{m}, c=63\ \text{m}, B=104^\circ$

Step1: Use SAS area formula

The formula for area with two sides and included angle is $Area=\frac{1}{2}ac\sin B$.

Step2: Substitute given values

$$Area=\frac{1}{2} \times 39 \times 63 \times \sin(104^\circ)$$
Calculate $\sin(104^\circ)\approx0.9703$, then:
$$Area=\frac{1}{2} \times 39 \times 63 \times 0.9703 = \frac{1}{2} \times 2457 \times 0.9703 \approx 1228.5 \times 0.9703 \approx 1192$$

---

Problem 20: $b=376\ \text{km}, c=538\ \text{km}, A=73^\circ$

Step1: Use SAS area formula

The formula for area with two sides and included angle is $Area=\frac{1}{2}bc\sin A$.

Step2: Substitute given values

$$Area=\frac{1}{2} \times 376 \times 538 \times \sin(73^\circ)$$
Calculate $\sin(73^\circ)\approx0.9563$, then:
$$Area=\frac{1}{2} \times 376 \times 538 \times 0.9563 = 188 \times 538 \times 0.9563 = 101144 \times 0.9563 \approx 96700$$

Answer:

1. $\boldsymbol{476\ \text{m}^2}$
2. $\boldsymbol{812\ \text{cm}^2}$
3. $\boldsymbol{870\ \text{ft}^2}$
4. $\boldsymbol{1192\ \text{m}^2}$
5. $\boldsymbol{96700\ \text{km}^2}$