Question

four points on a coordinate grid are a (2, 7), b (10, 3), c (5, 3) and d (11, 5).
a draw line segments ab and cd on a coordinate grid.
b point e is where line segments ab and cd cross. write the coordinates of e.
c show, using calculations, that e is the midpoint of cd.
d show, using calculations, that e lies \\( \frac { 3 } { 4 } \\) of the way along ab.

Explanation:

Step1: Find the equation of line \(AB\)

The slope of line \(AB\) with \(A(2,7)\) and \(B(10,3)\) is \(m_{AB}=\frac{y_{B}-y_{A}}{x_{B}-x_{A}}=\frac{3 - 7}{10 - 2}=\frac{-4}{8}=-\frac{1}{2}\).
Using the point - slope form \(y - y_{A}=m(x - x_{A})\), with \(A(2,7)\), we get \(y-7=-\frac{1}{2}(x - 2)\), which simplifies to \(y=-\frac{1}{2}x+8\).

Step2: Find the equation of line \(CD\)

The slope of line \(CD\) with \(C(5,3)\) and \(D(11,5)\) is \(m_{CD}=\frac{y_{D}-y_{C}}{x_{D}-x_{C}}=\frac{5 - 3}{11 - 5}=\frac{2}{6}=\frac{1}{3}\).
Using the point - slope form \(y - y_{C}=m(x - x_{C})\), with \(C(5,3)\), we get \(y - 3=\frac{1}{3}(x - 5)\), which simplifies to \(y=\frac{1}{3}x+\frac{4}{3}\).

Step3: Find the coordinates of \(E\) (intersection of \(AB\) and \(CD\))

Set \(-\frac{1}{2}x+8=\frac{1}{3}x+\frac{4}{3}\).
Multiply through by \(6\) to clear the fractions: \(-3x + 48 = 2x+8\).
Add \(3x\) to both sides: \(48=5x + 8\).
Subtract \(8\) from both sides: \(40 = 5x\), so \(x = 8\).
Substitute \(x = 8\) into \(y=-\frac{1}{2}x + 8\): \(y=-\frac{1}{2}(8)+8=4\). So \(E=(8,4)\).

Step4: Check if \(E\) is the mid - point of \(CD\)

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\).
For \(C(5,3)\) and \(D(11,5)\), the mid - point is \((\frac{5 + 11}{2},\frac{3+5}{2})=(8,4)\), so \(E\) is the mid - point of \(CD\).

Step5: Check if \(E\) is \(\frac{3}{4}\) of the way along \(AB\)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
The distance \(AB=\sqrt{(10 - 2)^2+(3 - 7)^2}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}\).
The distance \(AE=\sqrt{(8 - 2)^2+(4 - 7)^2}=\sqrt{36+9}=\sqrt{45}=3\sqrt{5}\).
\(\frac{AE}{AB}=\frac{3\sqrt{5}}{4\sqrt{5}}=\frac{3}{4}\)

Answer:

Coordinates of \(E=(8,4)\)