Question

1. (mangle c = 110^{circ},b = 35\text{ mi},c = 38\text{ mi}) 9) (b = 14\text{ cm},mangle a=107^{circ},c = 23\text{ cm}) 10) (mangle a = 115^{circ},c = 19\text{ mi},a = 28\text{ mi}) find the area of each triangle to the nearest tenth. 11) 12) 13) 14)

Explanation:

Step1: Recall the area formula for a triangle

The area formula for a triangle when two - sides and the included - angle are given is $A=\frac{1}{2}bc\sin A$ (or equivalent forms depending on the given sides and angle). When three - sides are given, we can use Heron's formula $A = \sqrt{s(s - a)(s - b)(s - c)}$, where $s=\frac{a + b + c}{2}$.

Step2: Solve problem 11

In triangle $RST$, $\angle R = 29^{\circ}$, $\angle T=109^{\circ}$, so $\angle S=180^{\circ}-(29^{\circ}+109^{\circ}) = 42^{\circ}$, and $RT = 16$ km. Using the formula $A=\frac{1}{2}ab\sin C$, with $a = 16$ km, and assuming we use the sides and the included - angle, say we use the side $RT$ and the angles. We can use $A=\frac{1}{2}(16)\times RS\times\sin T$. First, we find the area using $A=\frac{1}{2}(16)\times RS\times\sin109^{\circ}$. But we can also use the formula $A=\frac{1}{2}(16)\times RT\times\sin S$. So $A=\frac{1}{2}(16)(16)\sin42^{\circ}\approx\frac{1}{2}(16)(16)(0.6691)\approx85.6$ km².

Step3: Solve problem 12

Given $RT = 13$ cm, $ST = 10$ cm, and $\angle T = 23^{\circ}$. Using the formula $A=\frac{1}{2}ab\sin C$, where $a = 13$ cm, $b = 10$ cm, and $C = 23^{\circ}$. Then $A=\frac{1}{2}(13)(10)\sin23^{\circ}=\frac{1}{2}(13)(10)(0.3907)\approx25.4$ cm².

Step4: Solve problem 13

Given $YZ = 7$ m, $ZX = 14$ m, and $\angle Z = 26^{\circ}$. Using the formula $A=\frac{1}{2}ab\sin C$, where $a = 7$ m, $b = 14$ m, and $C = 26^{\circ}$. Then $A=\frac{1}{2}(7)(14)\sin26^{\circ}=\frac{1}{2}(7)(14)(0.4384)\approx21.5$ m².

Step5: Solve problem 14

Given $PQ = 11$ yd, $QR = 13$ yd, $PR = 13$ yd. First, find $s=\frac{11 + 13+13}{2}=\frac{37}{2}=18.5$ yd. Then use Heron's formula $A=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{18.5(18.5 - 11)(18.5 - 13)(18.5 - 13)}=\sqrt{18.5\times7.5\times5.5\times5.5}=\sqrt{18.5\times7.5\times30.25}\approx\sqrt{4145.3125}\approx64.4$ yd².

Answer:

1. $85.6$ km²
2. $25.4$ cm²
3. $21.5$ m²
4. $64.4$ yd²