Curriculum Alignment
- Strand
- 2.0 Measurements and Geometry
- Sub-Strand
- 2.9 Vectors 1
Subsection 2.9.4 Midpoints of Vectors
Teacher Resource 2.9.39.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 2.9.6.
(a)
(b)
Choose any starting point on the graph and label it as Point \(A\text{.}\) write down its coordinates.
(c)
From Point \(A\text{,}\) move \(6\) units to the right parallel to the \(x\) axis and mark this new location as Point \(B\text{.}\) Write down its coordinates.
(d)
(e)
(f)
Identify the coordinates of Point \(M\text{.}\)
(g)
Think of a way to determine coordinates of Point \(M\) without manually counting the units.
(h)
Discuss and share your findings with the rest of the class.
Key Takeaway 2.9.40.
Consider the cordinates of point \(P\) given as \((x_1,y_1)\) and point \(N\) given as \((x_2,y_2)\) and \(M\) is the midpoint of \(\mathbf{PN}\) as shown in figure below.
\begin{align*}
\mathbf{OM} \amp = \mathbf{OP} + \mathbf{PM} \\
\amp = \mathbf{a} + \frac{1}{2} \mathbf{PN}\\
\amp = \mathbf{a} + \frac{1}{2} (\mathbf{b} - \mathbf{a})\\
\amp = \mathbf{a} + \frac{1}{2}\mathbf{b} - \frac{1}{2}\mathbf{a}\\
\amp = \mathbf{a} - \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}\\
\amp = \frac{\mathbf{a} + \mathbf{b}}{2}
\end{align*}
But \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\)
Thus, midpoint \(\mathbf{M} = \left( \frac{ x_1 + x_2 }{ 2}, \frac{ y_1 + y_2 }{ 2}\right)\)
Example 2.9.42.
Find the coordinates of the midpoint \(M\) of \(\textbf{AB}\) given the following points \(A(6,1), \,\, B(4,3)\text{.}\)
Solution.
To determine the midpoint \(M\) of \(\overrightarrow{AB}\) , we apply the midpoint formulae as follows.
\begin{align*}
\text{Midpoint } \amp = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \\
\amp = \left(\frac{6 + 4}{2}, \frac{1+3}{2}\right)\\
\amp = \left(\frac{10}{2}, \frac{4}{2}\right)\\
\amp = \left(5, 2\right)
\end{align*}
Example 2.9.43.
The points \(PQR\) are on a straight line, given point \(Q(-3,5)\) is the midpoint of \(PR\text{.}\) Find the coordinates of \(P\) if \(R(-10,15)\text{?}\)
Exercises Exercises
1.
Find the coordinates of the midpoint of \(\mathbf{PQ}\) in each of the following cases:
-
\(\displaystyle P\, (-5,6), Q\, (3,-4)\)
-
\(\displaystyle P\,(1,−3), Q\,(5,7)\)
-
\(\displaystyle P\, (-2,4), Q\, (6,0)\)
-
\(\displaystyle P\, (a,b), Q\, (c,d)\)
Answer.
-
\(\displaystyle \left( \frac{-5+3}{2} ,\frac{6-4}{2}\right) = (-1,1)\)
-
\(\displaystyle \left( \frac{1+5}{2} ,\frac{-3+7}{2}\right) = (3,2)\)
-
\(\displaystyle \left( \frac{-2+6}{2} ,\frac{4+0}{2}\right) = (2,2)\)
-
\(\displaystyle \left( \frac{a+c}{2} ,\frac{b+d}{2}\right) = \left( \frac{a+c}{2} ,\frac{b+d}{2}\right)\)
2.
In the figure below, \(ABC\) is a triangle where \(M\) and \(N\) are midpoints of \(\overrightarrow{AC}\text{,}\) \(\overrightarrow{AB}\) respectively.
Triangle \(ABC\) has vertices at points \(A(2,2)\text{,}\) \(B(6,2)\) and \(C(4,6)\text{.}\) Determine the coordinates of \(M\) and \(N\text{.}\)
