Question

M is the midpoint of $\overline{AB}$, Q is the midpoint of $\overline{AM}$, and T is the midpoint of $\overline{QM}$. If the coordinates of A and B are a and b, find the coordinates of Q and T in terms of a and b.

Short answer

Coordinates of Q is $\frac{3a+b}{4}$ andTis$\frac{5a+3b}{8}$

Step-by-step solution

Step 1. State the definition of midpoint.

A midpoint is a point on a line segment that divides it into equal segments.

Here, M is the midpoint of $\overline{AB}$, Q is the midpoint of $\overline{AM}$, and T is the midpoint of $\overline{QM}$and the coordinates of A and B are a and b respectively.

Step 2. Calculate the coordinate of Q.

As M is the midpoint of $\overline{AB}$,

Then, the coordinate of AM is $\frac{a+b}{2}$.

It is also given that Q is the midpoint of $\overline{AM}$,

Then, the coordinate of Q is:

$\begin{array}{c}\frac{a+\frac{a+b}{2}}{2}=\frac{\frac{2a+a+b}{2}}{2}\\ =\frac{3a+b}{4}\end{array}$

Step 3. Calculate the coordinate of T.

As T is the midpoint of $\overline{QM}$,

Then, the coordinate of T is:

$\frac{\frac{a+b}{2}+\frac{3a+b}{4}}{2}=\frac{\frac{2a+2b+3a+b}{4}}{2}$

$=\frac{2a+2b+3a+b}{8}$

$=\frac{5a+3b}{8}$

Step 4. State the conclusion.

Therefore, the coordinate of Q is $\frac{3a+b}{4}$ and the coordinate of T is $\frac{5a+3b}{8}$.