Notes

 

Intro

An oblique triangle is any triangle that is not a right triangle. To find the area of these triangles you would use either Heron’s Formula or the Law of sines formula as shown above.

Questions

\(\textbf{1)}\) Find the area of this triangle.

Show Area

The area is \(= \displaystyle\frac{15\sqrt{3}}{4} \approx 6.4952 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(5)(3) \sin{(120)}\)
\(\,\,\,\,\,\,\text{Area}=\frac{15}{2} \cdot \frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\text{Area}= \displaystyle\frac{15\sqrt{3}}{4} \approx 6.4952\) units squared

 

\(\textbf{2)}\) Find the area of this triangle.

Show Answer

\(\text{The area is }\approx 19.90\)

Show Work

\(\text{Step 1: Find Semiperimeter}\)
\(\,\,\,\,\,\,s=\displaystyle \frac{a+b+c}{2}\)
\(\,\,\,\,\,\,s=\displaystyle \frac{5+8+9}{2}\)
\(\,\,\,\,\,\,s=\displaystyle 11\)
\(\text{Step 2: Find the Area}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{11(11-9)(11-8)(11-5)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{11(2)(3)(6)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{396}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle 6\sqrt{11}\)
\(\text{The area is }\approx 19.90\) units squared

 

\(\textbf{3)}\) Find the area of this triangle.

Show Area

The area is \(\approx 10.5497 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(6)(9) \sin{(23)}\)
\(\,\,\,\,\,\,\text{Area}=27 \cdot \left(0.39073\right)\)
\(\,\,\,\,\,\,\text{Area}\approx 10.5497\) units squared

 

\(\textbf{4)}\) Find the area of this triangle.

Show Answer

The area is \(\approx13.42\) units squared

Show Work

\(\text{Step 1: Find Semiperimeter}\)
\(\,\,\,\,\,\,s=\displaystyle \frac{a+b+c}{2}\)
\(\,\,\,\,\,\,s=\displaystyle \frac{9+7+4}{2}\)
\(\,\,\,\,\,\,s=\displaystyle 10\)
\(\text{Step 2: Find the Area}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{10(10-9)(10-7)(10-4)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{10(1)(3)(6)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{180}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle 3\sqrt{20}\)
\(\text{The area is }\approx13.42\) units squared

 

\(\textbf{5)}\) Find the area of this triangle.

Show Area

The area is \(\approx 11.2763 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(3)(8) \sin{(70)}\)
\(\,\,\,\,\,\,\text{Area}=12 \cdot \left(0.93969\right)\)
\(\,\,\,\,\,\,\text{Area}\approx 11.2763\) units squared

 

\(\textbf{6)}\) Find the area of this triangle.

Show Answer

\(\text{The area is }\approx 35.14\)

Show Work

\(\text{Step 1: Find Semiperimeter}\)
\(\,\,\,\,\,\,s=\displaystyle \frac{a+b+c}{2}\)
\(\,\,\,\,\,\,s=\displaystyle \frac{6+14+18}{2}\)
\(\,\,\,\,\,\,s=\displaystyle 19\)
\(\text{Step 2: Find the Area}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{19(19-6)(19-14)(19-18)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{18(13)(5)(1)}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{1170}\)
\(\,\,\,\,\,\,\text{Area}=\displaystyle 3\sqrt{130}\)
\(\text{The area is }\approx 35.14\) units squared

 

\(\textbf{7)}\) What is the area of this triangle?

Show Answer

The answer is \(\approx 5.2\) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(4)(3) \sin{(60)}\)
\(\,\,\,\,\,\,\text{Area}=6 \cdot \frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\)The answer is \(\approx 5.2\) units squared

 

\(\textbf{8)}\) Find the area of this triangle.

Show Area

The area is \(= \displaystyle \frac{3\sqrt{3}}{4} \approx 1.2990 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(1)(3) \sin{(60)}\)
\(\,\,\,\,\,\,\text{Area}=\frac{3}{2} \cdot \frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\text{Area}= \displaystyle \frac{3\sqrt{3}}{4} \approx 1.2990\) units squared

 

 

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