Answer

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**Hint:**In this problem, we have to find the value of k if \[AB\bot CD\]. Here we can see that we are given some points, with which we can find the value of slope from the two points formula,\[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]. We will get two values of slope. We are also given that they are perpendicular, as we know that if two slope are perpendicular, then we will have the condition \[{{m}_{1}}\times {{m}_{2}}=-1\], by using this condition we can find the value of k.

**Complete step by step answer:**

Here we have to find the value of k, if \[AB\bot CD\].

We know that the given points are,

\[A\left( 0,9 \right),B\left( 1,11 \right),C\left( 3,13 \right),D\left( 7,k \right)\]

We can now find the value of slope from the two points formula,\[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\].

We can now find the slope value for \[A\left( 0,9 \right),B\left( 1,11 \right)\]

Slope of AB,

\[\Rightarrow {{m}_{1}}=\dfrac{11-9}{1-0}=2\]

Slope of AB, \[{{m}_{1}}=2\]……. (1)

We can now find the slope of \[C\left( 3,13 \right),D\left( 7,k \right)\]

Slope of CD,

\[\Rightarrow {{m}_{2}}=\dfrac{k-13}{7-3}=\dfrac{k-13}{4}\]

Slope of CD, \[{{m}_{2}}=\dfrac{k-13}{4}\]……… (2)

We know that if two slopes are perpendicular, then we will have the condition \[{{m}_{1}}\times {{m}_{2}}=-1\].

As we have \[AB\bot CD\] we can substitute (1) and (2) in the above condition, we get

\[\Rightarrow 2\times \dfrac{k-13}{4}=-1\]

We can now simplify the above step, we get

\[\begin{align}

& \Rightarrow \dfrac{k-13}{2}=-1 \\

& \Rightarrow k-13=-2 \\

& \Rightarrow k=13-2=11 \\

\end{align}\]

Therefore, the value of k is 11.

**Note:**We should always remember that if two slope are perpendicular, then we will have the condition \[{{m}_{1}}\times {{m}_{2}}=-1\], where we can find the value of slope as we are given four points, with the two points form \[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]. We should substitute the value of x and y correctly to get the value of the slope.

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