Two sides AB and BC and median AM of one tria | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(N/A) In $\Delta ABC$,$AM$ is a median [Given].$	herefore BM = \frac{1}{2} BC$ ........... $(1)$In $\Delta PQR$,$PN$ is a median.$	herefore QN = \fr

Vedclass · Accepted Answer

Vedclass · Answer

(N/A) In $\Delta ABC$,$AM$ is a median [Given].
$\therefore BM = \frac{1}{2} BC$ ........... $(1)$
In $\Delta PQR$,$PN$ is a median.
$\therefore QN = \frac{1}{2} QR$ ........... $(2)$
$\because BC = QR$ [Given]
$\therefore \frac{1}{2} BC = \frac{1}{2} QR$
$\Rightarrow BM = QN$ [From $(1)$ and $(2)$]
$(i)$ In $\Delta ABM$ and $\Delta PQN$,we have:
$AB = PQ$ [Given]
$AM = PN$ [Given]
$BM = QN$ [Proved above]
$\therefore \Delta ABM \cong \Delta PQN$ [$SSS$ congruence criterion]
$(ii)$ $\because \Delta ABM \cong \Delta PQN$
$\therefore$ Their corresponding parts are congruent $(CPCT)$.
$\Rightarrow \angle B = \angle Q$
Now,in $\Delta ABC$ and $\Delta PQR$,we have:
$AB = PQ$ [Given]
$\angle B = \angle Q$ [Proved above]
$BC = QR$ [Given]
$\therefore \Delta ABC \cong \Delta PQR$ [$SAS$ congruence criterion]

Two sides $AB$ and $BC$ and median $AM$ of one triangle $ABC$ are respectively equal to sides $PQ$ and $QR$ and median $PN$ of $\Delta PQR$ (see Fig). Show that:
$(i)$ $\Delta ABM \cong \Delta PQN$
$(ii)$ $\Delta ABC \cong \Delta PQR$

Similar Questions

$E$ and $F$ are respectively the mid-points of equal sides $AB$ and $AC$ of $\Delta ABC$ (see figure). Show that $BF = CE$.

In the given figure,$AB \perp BQ$,$PQ \perp QB$,$AC = PR$ and $BR = QC$. Prove that $\angle BAC = \angle QPR$.

$ABCD$ is a quadrilateral in which $AB = BC$ and $AD = CD$. Show that $BD$ bisects both the angles $ABC$ and $ADC$.

$AB$ is a line segment. $P$ and $Q$ are points on opposite sides of $AB$ such that each of them is equidistant from the points $A$ and $B$ (see figure). Show that the line $PQ$ is the perpendicular bisector of $AB$.

$l$ and $m$ are two parallel lines intersected by another pair of parallel lines $p$ and $q$ (see Fig). Show that $\Delta ABC \cong \Delta CDA$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Two sides $AB$ and $BC$ and median $AM$ of one triangle $ABC$ are respectively equal to sides $PQ$ and $QR$ and median $PN$ of $\Delta PQR$ (see Fig). Show that:$(i)$ $\Delta ABM \cong \Delta PQN$$(ii)$ $\Delta ABC \cong \Delta PQR$