Question
Prove that the line segment joining the midpoints of two sides of a triangle is parallel to and half of the third side.
✓
Answer
Let the triangle be $ABC$. If $M$ and $N$ are the midpoints of $AB$ and $AC$ respectively, then $\overrightarrow{ AM }=1 / 2 \overrightarrow{ AB }$ and $\overrightarrow{ AN }=1 / 2 \overrightarrow{ AC }$.
Thus by triangle law $ \overrightarrow{ AN }=\overrightarrow{ AM }+\overrightarrow{ MN }$
$\therefore \overline{ MN }=\overrightarrow{ AN }-\overline{ AM }=\frac{\overrightarrow{ AC }-\overline{ AB }}{2}=\frac{\overrightarrow{ BC }}{2} $
Thus, $\overrightarrow{ MN }$ is parallel to and half as long as $\overline{ BC }$.
Thus by triangle law $ \overrightarrow{ AN }=\overrightarrow{ AM }+\overrightarrow{ MN }$
$\therefore \overline{ MN }=\overrightarrow{ AN }-\overline{ AM }=\frac{\overrightarrow{ AC }-\overline{ AB }}{2}=\frac{\overrightarrow{ BC }}{2} $
Thus, $\overrightarrow{ MN }$ is parallel to and half as long as $\overline{ BC }$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Evaluate:
→$\int \frac{x-2}{\sqrt{x+5}} \cdot d x$
Integrate the following w.r.t. x:
→$\log \left(x^2+1\right)$
Evaluate the following integrals:
$\int\frac{\text{e}^\text{x}}{(1+\text{e}^\text{x})^2}\text{dx}$
→$\int\frac{\text{e}^\text{x}}{(1+\text{e}^\text{x})^2}\text{dx}$
If from a point $Q (a, b, c)$ perpendiculars $QA$ and $QB$ are drawn to the $YZ$ and $ZX$ planes respectively, then find the vector equation of the plane $QA B$.
→Construct the truth table for each of the following : [(~p ∨ q) ∧ (q → r)] → (p → r)
A man of height 2 metres walks at a uniform speed of 6 km/hr away from a lamp post of 6 metres high. Find the rate at which the length of the shadow is increasing.
If matrix $A = [1 2 3]$, write $AA^T$.
→Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}8&2&7\\12&3&5\\16&4&3 \end{vmatrix}$
→$\begin{vmatrix}8&2&7\\12&3&5\\16&4&3 \end{vmatrix}$
Check the ordered points (1, −1), (2, −1) is a solution of 2x + 3y − 6 ≤ 0
The radius of a circle is increasing at the rate of 0.5cm/ sec. Find the rate of increase of its circumference.