Question

for help with question 4, see example 3. 4. find the length of the median from vertex k.

Explanation:

Step1: Find the mid - point of the opposite side

The opposite side of vertex \(K(4,0)\) is the line segment \(IL\). The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For \(I(1,2)\) and \(L(3,4)\), \(x_1 = 1,y_1=2,x_2 = 3,y_2 = 4\). Then the mid - point \(M\) of \(IL\) is \((\frac{1 + 3}{2},\frac{2+4}{2})=(2,3)\).

Step2: Calculate the distance between \(K(4,0)\) and the mid - point \((2,3)\)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 4,y_1 = 0,x_2=2,y_2 = 3\). So \(d=\sqrt{(2 - 4)^2+(3 - 0)^2}=\sqrt{(-2)^2+3^2}=\sqrt{4 + 9}=\sqrt{13}\).

Answer:

\(\sqrt{13}\)