Question

select the correct answer from each drop - down menu.
in the diagram, \\( \overline { a b } \\) is divided into equal parts. the coordinates of point \\( a \\) are \\( ( - 3,9 ) \\), and the coordinates of point \\( b \\) are \\( ( 9,5 ) \\).
the coordinates of point \\( c \\) are
the coordinates of point \\( e \\) are
the coordinates of point \\( h \\) are

Explanation:

Step1: Calculate the difference in x - coordinates and y - coordinates

The formula for the mid - point between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((x_1+\frac{n}{m}(x_2 - x_1),y_1+\frac{n}{m}(y_2 - y_1))\), where \(m\) is the total number of equal parts and \(n\) is the number of parts from the first point. Here, \(A(-3,9)\) and \(B(9,5)\), and \(AB\) is divided into 8 equal parts.
For the \(x\) - coordinate difference: \(\Delta x=x_B - x_A=9-(-3)=12\). For the \(y\) - coordinate difference: \(\Delta y=y_B - y_A=5 - 9=-4\).

Step2: Calculate the coordinates of point \(C\)

Since \(C\) is 1 part away from \(A\) (\(n = 1,m = 8\)).
\(x_C=-3+\frac{1}{8}\times12=-3 + 1.5=-1.5\)
\(y_C=9+\frac{1}{8}\times(-4)=9-0.5 = 8.5\)

Step3: Calculate the coordinates of point \(E\)

Since \(E\) is 3 parts away from \(A\) (\(n = 3,m = 8\)).
\(x_E=-3+\frac{3}{8}\times12=-3 + 4.5 = 1.5\)
\(y_E=9+\frac{3}{8}\times(-4)=9-1.5=7.5\)

Step4: Calculate the coordinates of point \(H\)

Since \(H\) is 6 parts away from \(A\) (\(n = 6,m = 8\)).
\(x_H=-3+\frac{6}{8}\times12=-3 + 9=6\)
\(y_H=9+\frac{6}{8}\times(-4)=9 - 3=6\)

Answer:

The coordinates of point \(C\) are \((-1.5,8.5)\). The coordinates of point \(E\) are \((1.5,7.5)\). The coordinates of point \(H\) are \((6,6)\).