Question

In a $\Delta ABC$ if C = 90 degree , then check whether $\dfrac{c}{r}=\dfrac{c+a}{b}+\dfrac{c+b}{a}$ is true or not

Answer 1

Hint: First we will draw the triangle and then we are going to use the sine formula to find the relation between the sides and the radius r. After using that we will try rearrange the following terms to check that $\dfrac{c}{r}=\dfrac{c+a}{b}+\dfrac{c+b}{a}$ is true or false.

Complete step-by-step answer:
Let’s start solving the question,

As we have drawn the figure now we will state the sine formula,
$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=\dfrac{1}{2r}$
We know that C = 90 degree, so we can put the value of C in the above equation,
From the triangle angle A + B = 90,
Therefore,
B = 90 – A
Now substituting the value of a, b, c in $\dfrac{c}{r}=\dfrac{c+a}{b}+\dfrac{c+b}{a}$ as,
$\begin{aligned} & a=2r\sin A \\\ & b=2r\sin B \\\ & c=2r \\\ \end{aligned}$
After putting the values we get,
LHS = $\dfrac{c}{r}=\dfrac{2r}{r}=2$
Now the value of RHS is:
$\begin{aligned} & =\dfrac{2r+2r\sin A}{2r\sin B}+\dfrac{2r+2r\sin B}{2r\sin A} \\\ & =\dfrac{1+\sin A}{\sin B}+\dfrac{1+\sin B}{\sin A} \\\ \end{aligned}$
Now substituting the value of B as 90 – A we get,
$\begin{aligned} & =\dfrac{1+\sin A}{\sin \left( 90-A \right)}+\dfrac{1+\sin \left( 90-A \right)}{\sin A} \\\ & =\dfrac{1+\sin A}{\cos A}+\dfrac{1+\cos A}{\sin A} \\\ & =\dfrac{\sin A+{{\sin }^{2}}A+\cos A+{{\cos }^{2}}A}{\sin A\cos A} \\\ \end{aligned}$
Now let’s put the value of A = 45 degree, and if it doesn’t give 2 then $\dfrac{c}{r}=\dfrac{c+a}{b}+\dfrac{c+b}{a}$ is false.
Putting the value of A = 45 we get,
$\begin{aligned} & =\dfrac{\dfrac{1}{\sqrt{2}}+{{\left( \dfrac{1}{\sqrt{2}} \right)}^{2}}+\dfrac{1}{\sqrt{2}}+{{\left( \dfrac{1}{\sqrt{2}} \right)}^{2}}}{\dfrac{1}{\sqrt{2}}\times \dfrac{1}{\sqrt{2}}} \\\ & =\dfrac{1+\sqrt{2}}{\dfrac{1}{2}} \\\ & =2\left( 1+\sqrt{2} \right) \\\ \end{aligned}$
Hence, it is not equal to 2.
Therefore, $\dfrac{c}{r}=\dfrac{c+a}{b}+\dfrac{c+b}{a}$ is false.
Note: We have used the sine formula in a triangle which is very useful and gives us the relation between sides, angle and r. Instead of solving the whole equation one should put any value of A and show that it is false. To prove a statement false we need just one example.