Class 9MathematicsAreas of Parallelograms and TrianglesTextbook - Areas of Parallelograms and Triangles
Medium$P$ and $Q$ are respectively the mid-points of sides $AB$ and $BC$ of a triangle $ABC$ and $R$ is the mid-point of $AP$. Show that $\text{ar} (PBQ) = \text{ar} (ARC)$.
(N/A) Given: $P$ is the mid-point of $AB$,$Q$ is the mid-point of $BC$,and $R$ is the mid-point of $AP$.
$1$. In $\Delta ABC$,$AQ$ is a median (since $Q$ is the mid-point of $BC$). Therefore,$\text{ar}(\Delta ABQ) = \frac{1}{2} \text{ar}(\Delta ABC)$.
$2$. In $\Delta ABQ$,$PQ$ is a median (since $P$ is the mid-point of $AB$). Therefore,$\text{ar}(\Delta PBQ) = \frac{1}{2} \text{ar}(\Delta ABQ) = \frac{1}{2} \times (\frac{1}{2} \text{ar}(\Delta ABC)) = \frac{1}{4} \text{ar}(\Delta ABC)$.
$3$. In $\Delta ABC$,$AC$ is a side. Consider $\Delta ARC$. Since $R$ is the mid-point of $AP$ and $P$ is the mid-point of $AB$,$AR = RP = PB = \frac{1}{4} AB$.
$4$. Alternatively,using the property of medians: In $\Delta APC$,$CR$ is a median. Thus,$\text{ar}(\Delta ARC) = \frac{1}{2} \text{ar}(\Delta APC)$.
$5$. Since $P$ is the mid-point of $AB$,$\text{ar}(\Delta APC) = \frac{1}{2} \text{ar}(\Delta ABC)$.
$6$. Therefore,$\text{ar}(\Delta ARC) = \frac{1}{2} \times (\frac{1}{2} \text{ar}(\Delta ABC)) = \frac{1}{4} \text{ar}(\Delta ABC)$.
Thus,$\text{ar}(\Delta PBQ) = \text{ar}(\Delta ARC) = \frac{1}{4} \text{ar}(\Delta ABC)$.
$1$. In $\Delta ABC$,$AQ$ is a median (since $Q$ is the mid-point of $BC$). Therefore,$\text{ar}(\Delta ABQ) = \frac{1}{2} \text{ar}(\Delta ABC)$.
$2$. In $\Delta ABQ$,$PQ$ is a median (since $P$ is the mid-point of $AB$). Therefore,$\text{ar}(\Delta PBQ) = \frac{1}{2} \text{ar}(\Delta ABQ) = \frac{1}{2} \times (\frac{1}{2} \text{ar}(\Delta ABC)) = \frac{1}{4} \text{ar}(\Delta ABC)$.
$3$. In $\Delta ABC$,$AC$ is a side. Consider $\Delta ARC$. Since $R$ is the mid-point of $AP$ and $P$ is the mid-point of $AB$,$AR = RP = PB = \frac{1}{4} AB$.
$4$. Alternatively,using the property of medians: In $\Delta APC$,$CR$ is a median. Thus,$\text{ar}(\Delta ARC) = \frac{1}{2} \text{ar}(\Delta APC)$.
$5$. Since $P$ is the mid-point of $AB$,$\text{ar}(\Delta APC) = \frac{1}{2} \text{ar}(\Delta ABC)$.
$6$. Therefore,$\text{ar}(\Delta ARC) = \frac{1}{2} \times (\frac{1}{2} \text{ar}(\Delta ABC)) = \frac{1}{4} \text{ar}(\Delta ABC)$.
Thus,$\text{ar}(\Delta PBQ) = \text{ar}(\Delta ARC) = \frac{1}{4} \text{ar}(\Delta ABC)$.
Explore More
Similar Questions
In the figure,$ABC$ and $ABD$ are two triangles on the same base $AB$. If line segment $CD$ is bisected by $AB$ at $O$,show that $\operatorname{ar}(ABC) = \operatorname{ar}(ABD)$.
Medium
View SolutionIn the figure,$P$ is a point in the interior of a parallelogram $ABCD$. Show that:
$(i)$ $\operatorname{ar}(APB) + \operatorname{ar}(PCD) = \frac{1}{2} \operatorname{ar}(ABCD)$
$(ii)$ $\operatorname{ar}(APD) + \operatorname{ar}(PBC) = \operatorname{ar}(APB) + \operatorname{ar}(PCD)$
[Hint: Through $P$,draw a line parallel to $AB$.]
$(i)$ $\operatorname{ar}(APB) + \operatorname{ar}(PCD) = \frac{1}{2} \operatorname{ar}(ABCD)$
$(ii)$ $\operatorname{ar}(APD) + \operatorname{ar}(PBC) = \operatorname{ar}(APB) + \operatorname{ar}(PCD)$
[Hint: Through $P$,draw a line parallel to $AB$.]
Difficult
View Solution$D, E$ and $F$ are respectively the mid-points of the sides $BC, CA$ and $AB$ of a $\Delta ABC$. Show that $BDEF$ is a parallelogram.
Medium
View SolutionIn the figure,diagonals $AC$ and $BD$ of quadrilateral $ABCD$ intersect at $O$ such that $OB = OD$. If $AB = CD$,then show that:
$(i)$ $ar(DOC) = ar(AOB)$
$(ii)$ $ar(DCB) = ar(ACB)$
$(iii)$ $DA \parallel CB$ or $ABCD$ is a parallelogram.
[Hint: From $D$ and $B$,draw perpendiculars to $AC$.]
$(i)$ $ar(DOC) = ar(AOB)$
$(ii)$ $ar(DCB) = ar(ACB)$
$(iii)$ $DA \parallel CB$ or $ABCD$ is a parallelogram.
[Hint: From $D$ and $B$,draw perpendiculars to $AC$.]
Difficult
View SolutionShow that a median of a triangle divides it into two triangles of equal areas.
Medium
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo