Three vertices of a triangle are (2,7),(4,-1),(0,Y).if the perimeter of the triangle is the least then express Y in the lowest rational form as (p/q).

Question

I don't know how to proceed in this question. What I know is that the perimeter is the sum of all the sides.

Answer 1

In this question we have a triangle formed by two points $A$ and $B$ which are fixed and a third point $C$ which you can move on the Y-axis.

To minimise the perimeter we have to only minimise the sum of the two sides $AC$ and $BC$. Reflect the point $A$ about the Y-axis and say the new point is $A'$. By symmetry we can say that $AC=A'C$. So now we gotta minimise the sum $AC+BC=A'C+BC$.

In the triangle $A'BC$ we have $$A'C+BC\ge A'B$$ Thus the sum will be minimum when the points $A'$,$C$ and $B$ lie on the same line. Now applying this condition in the coordinate system,$$\frac{y-7}{0+2}=\frac{-1-7}{6}\implies y=\frac{13}3$$

Answer 2

Notice, the length of side with vertices $(2, 7)$ & $(4, -1)$, using distance formula, is $$=\sqrt{(2-4)^2+(7+1)^2}=\sqrt{68}$$ Similarly, the length of side with vertices $(2, 7)$ & $(0, y)$ is $$=\sqrt{(2-0)^2+(7-y)^2}=\sqrt{(y-7)^2+4}$$ the length of side with vertices $(4, -1)$ & $(0, y)$ is $$=\sqrt{(4-0)^2+(-1-y)^2}=\sqrt{(y+1)^2+16}$$ Hence, the perimeter $P$ is the sum of all three sides hence. we have $$P=\sqrt{68}+\sqrt{(y-7)^2+4}+\sqrt{(y+1)^2+16}$$ Differentiating w.r.t. $y$, we get $$\frac{dP}{dy}=\frac{y-7}{\sqrt{(y-7)^2+4}}+\frac{y+1}{\sqrt{(y+1)^2+16}}$$ $$\frac{d^2P}{dy^2}=\frac{4}{((y-7)^2+4)^{3/2}}+\frac{16}{((y-7)^2+4)^{3/2}}$$ It is clear that $\frac{d^2P}{dy^2}>0\ \ \forall\ \ y\in R$ Hence, the perimeter is minimum/least

Now, setting $\frac{dP}{dy}=0$ for minimum, we get $$\frac{y-7}{\sqrt{(y-7)^2+4}}+\frac{y+1}{\sqrt{(y+1)^2+16}}=0$$ $$(y+1)^2=4(y-7)^2$$ $$y+1=\pm 2(y-7)$$

$$\implies y=15, \ \ \frac{13}{3} $$ But, it is clear that at both the values $y=15$ & $y=\frac{13}{5}$, the perimeter $P$ of triangle is minimum/least.

But at $y=15$ the triangle does not exist hence $y=15$ is unacceptable.

Hence, we get $$\color{red}{y=\frac{13}{3}} $$