$E$ and $F$ are respectively the mid-points of equal sides $AB$ and $AC$ of $\Delta ABC$ (see figure). Show that $BF = CE$.
(N/A) In $\Delta ABF$ and $\Delta ACE$:
$AB = AC$ (Given)
$\angle A = \angle A$ (Common angle)
$AF = AE$ (Since $F$ and $E$ are mid-points of equal sides $AC$ and $AB$ respectively,$AF = \frac{1}{2} AC$ and $AE = \frac{1}{2} AB$. Since $AB = AC$,it follows that $AF = AE$.)
Therefore,$\Delta ABF \cong \Delta ACE$ by the $SAS$ congruence rule.
Since the triangles are congruent,their corresponding parts are equal.
Thus,$BF = CE$ (by $CPCT$).
$AB = AC$ (Given)
$\angle A = \angle A$ (Common angle)
$AF = AE$ (Since $F$ and $E$ are mid-points of equal sides $AC$ and $AB$ respectively,$AF = \frac{1}{2} AC$ and $AE = \frac{1}{2} AB$. Since $AB = AC$,it follows that $AF = AE$.)
Therefore,$\Delta ABF \cong \Delta ACE$ by the $SAS$ congruence rule.
Since the triangles are congruent,their corresponding parts are equal.
Thus,$BF = CE$ (by $CPCT$).
Explore More
Similar Questions
In the figure,$PR > PQ$ and $PS$ bisects $\angle QPR$. Prove that $\angle PSR > \angle PSQ$.
Difficult
View Solution$P$ is a point equidistant from two lines $l$ and $m$ intersecting at point $A$. Show that the line $AP$ bisects the angle between them.
Medium
View Solution$\Delta ABC$ is an isosceles triangle in which $AB = AC$. Side $BA$ is produced to $D$ such that $AD = AB$ (see figure). Show that $\angle BCD$ is a right angle.
Difficult
View SolutionIn the figure,$OA = OB$ and $OD = OC$. Show that
$(i)$ $\Delta AOD \cong \Delta BOC$ and
$(ii)$ $AD \parallel BC$.
$(i)$ $\Delta AOD \cong \Delta BOC$ and
$(ii)$ $AD \parallel BC$.
Medium
View SolutionIn $\Delta ABC$,$AD$ is the perpendicular bisector of $BC$ (see figure). Show that $\Delta ABC$ is an isosceles triangle in which $AB = AC$.
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