Area of a Triangle using Radius and Perimeter
The area of a triangle can be calculated using its perimeter and the radius of the circle inscribed in the triangle.
Worked Examples of the Area of a Triangle using Radius and Perimeter
(a) Find an expression for the area of \(\triangle LOM \).
\( \begin{aligned} \displaystyle
\text{Area of } \triangle LOM &= \frac{1}{2} \times \text{base} \times \text{height} \\
&= \frac{1}{2} kr \\
\end{aligned} \\ \)
(b) Show that the area of \( \triangle KLM \) is given by \( \displaystyle A = \frac{1}{2} Tr \), where \(T\) is the perimeter of \( \triangle KLM \).
\( \begin{aligned} \require{AMSsymbols} \require{color} \displaystyle
A &= \frac{1}{2} kr + \frac{1}{2} \ell r + \frac{1}{2} mr \\
&= \frac{1}{2} (k+\ell +m)r \\
\therefore A &= \frac{1}{2} Tr &\color{red} k\color{red}+\color{red}\ell \color{red}+\color{red}m \color{red}= \color{red}T \\
\end{aligned} \\ \)
(c) Find how far from the foot of the fence the board touches the ground.
\( \text{Let the base of the triangle by } 2+b. \)
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
\text{Area of the triangle using its base and height } &= \frac{1}{2} \times 8 \times (2+b) \color{red} \cdots (1) \\
\text{Area of the triangle using } \frac{1}{2} Tr &= \frac{1}{2} \times (6+6+2+2+b+b) \times 2 \color{red} \cdots (2) \\
\frac{1}{2} \times 8 \times (2+b) &= \frac{1}{2} \times (6+6+2+2+b+b) \times 2 &\color{red} (1) = (2) \\
16 + 8b &= 32 + 4b \\
4b &= 16 \\
b &= 4 \\
2+b &= 6
\end{aligned} \)
\( \text{Therefore the board touches the ground at a point 6 m from the fence.} \)
(d) Find the radius of the second circle.
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
MP &= \sqrt{8^2 + 6^2} \\
&= 10 \\
MQ &= \sqrt{8^2 + (6+9)^2} \\
&= 17 \\
\text{Area of } \triangle MPQ \text{ using its base and height } &= \frac{1}{2} \times 9 \times 8 \color{red} \cdots (3) \\
\text{Area of } \triangle MPQ \text{ using } \frac{1}{2} Tr &= \frac{1}{2} \times (10+17+9) \times r \color{red} \cdots (4) \\
\frac{1}{2} \times (10+17+9) \times r &= \frac{1}{2} \times 9 \times 8 &\color{red} (4) = (3) \\
18 r &= 36 \\
\therefore r &= 2
\end{aligned} \)
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