From the given figure if \(AB = AC\), and \(P\), \(Q\) and \(R\) are mid-points of \(AC\), \(AB\), and \(BC\) respectively. Prove that \(PR = QR\).
Proof:
In \(\triangle ABC\), \(AB = AC\).
Thus, \(\triangle ABC\) is an triangle.
We know that, 'Angles opposite to equal sides of an isosceles triangle are equal.'
Therefore, the angles opposite to sides \(AB\) and \(AC\) are equal.
Thus, \(\angle ABC = \angle ACB \longrightarrow (1)\)
Therefore, the angles opposite to sides \(AB\) and \(AC\) are equal.
Thus, \(\angle ABC = \angle ACB \longrightarrow (1)\)
It is also known that \(P\), \(Q\) and \(R\) are mid-points of , , and respectively.
Mid-points divide the side into halves.
The two halves of \(AB\) are \(AQ\) and \(QB\). Similarly, the two halves of \(AC\) are \(AP\) and \(PC\).
Since \(AB = AC\), \(AQ = QB = AP = PC \longrightarrow (2)\)
In order to prove that \(PR = QR\), we should prove that the triangles \(CPR\) and \(QBR\) are congruent to each other.
We know from \((2)\) that \(QB = PC\).
Also, \(R\) is the mid-point of \(BC\).
Therefore, \(BR = RC\).
Also from \((1)\), \(\angle ABC = \angle ACB\)
Since \(AB = AC\), \(AQ = QB = AP = PC \longrightarrow (2)\)
In order to prove that \(PR = QR\), we should prove that the triangles \(CPR\) and \(QBR\) are congruent to each other.
We know from \((2)\) that \(QB = PC\).
Also, \(R\) is the mid-point of \(BC\).
Therefore, \(BR = RC\).
Also from \((1)\), \(\angle ABC = \angle ACB\)
Here, two pairs of corresponding sides and one pair of corresponding angles are equal.
Thus by congruence criterion, \(\triangle CPR \cong \triangle QBR\)
Thus by congruence criterion, \(\triangle CPR \cong \triangle QBR\)
Since \(\triangle CPR \cong \triangle QBR\), and by , \(PR = QR\).