Question

o is the point (0, 0) and a is the point (5, 8). points a and b are equally spaced along the same line such that the distance oa is equal to the distance ab.
a what are the coordinates of point b?
b c is the next point along the same line such that distance bc is equal to distances oa and ab. what are the coordinates of point c?
c the points continue along the line, equally spaced. each point is labelled with a letter of the alphabet, in order from a to z. show that point e has coordinates (25, 40).
d what are the coordinates of point t? show how you worked out your answer.
e write an expression for the coordinates of the nth letter of the alphabet along this same line.

Explanation:

Step1: Find the coordinates of point B

Since OA and AB are equal in length.
The coordinates of \(O=(0,0)\) and \(A=(5,8)\).
Using the mid - point formula (but here since \(OA = AB\), if we consider the vector \(\overrightarrow{OA}=(5,8)\), then \(\overrightarrow{AB}=\overrightarrow{OA}\).
If \(A=(x_1,y_1)=(5,8)\) and \(\overrightarrow{AB}=(5,8)\), then the coordinates of \(B=(x_1 + 5,y_1+8)=(10,16)\)

Step2: Find the coordinates of point C

Since \(BC = AB\) and \(B=(10,16)\), \(\overrightarrow{BC}=\overrightarrow{AB}=(5,8)\)
The coordinates of \(C=(x_2 + 5,y_2 + 8)\) where \(x_2 = 10,y_2=16\)
So \(C=(15,24)\)

Step3: General formula for the \(n\)th point

We observe that for the \(n\)th point (starting from \(A\) as the first point, \(n = 1\) corresponds to \(A=(5,8)\), \(n = 2\) corresponds to \(B=(10,16)\), \(n=3\) corresponds to \(C=(15,24)\))
The \(x\) - coordinate \(x_n=5n\) and the \(y\) - coordinate \(y_n = 8n\)

Answer:

a. \((10,16)\)
b. \((15,24)\)
c. Let \(A=(5,8)\), \(B=(10,16)\), \(C=(15,24)\)
We know that the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For \(A=(5,8)\) and \(E=(25,40)\)
\(d_{AE}=\sqrt{(25 - 5)^2+(40 - 8)^2}=\sqrt{20^2+32^2}=\sqrt{400 + 1024}=\sqrt{1424}\)
For \(A=(5,8)\) and \(B=(10,16)\): \(d_{AB}=\sqrt{(10 - 5)^2+(16 - 8)^2}=\sqrt{25 + 64}=\sqrt{89}\)
Since \(25=5\times5\) and \(40 = 8\times5\), the point \(E\) is the 5th point after \(A\) (counting \(A\) as the first point).
We can also use the vector method. The vector \(\overrightarrow{OA}=(5,8)\), if a point \(P=(x,y)\) is on the line, then \(\overrightarrow{OP}=k\overrightarrow{OA}\) where \(k\) is a scalar. For \(E=(25,40)\), \(k = 5\)

d. The coordinates of \(T\):
Since the points are of the form \((5n,8n)\)
e. The coordinates of the \(n\)th letter (assuming \(A\) is the first letter, \(n = 1\)) is \((5n,8n)\)