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Show that

i) D is the mid-point of AC

ii) MD is perpendicular to AC

iii) CM = MA = (\(\frac{1}{2} \)) AB

Answer :

Given: \(\triangle{ABC}\) is a right angled triangle \(\Rightarrow \) \(\angle{C}\) = \(90^\circ\)

and M is the mid-point of AB.

Also, DM || BC

i) In \(\triangle{ABC}\), M is the mid-point of AB and BC || MD,

So, By converse of mid-point theorem,

D is the mid-point of AC.

\(\Rightarrow \) AD = CD ...(i)

ii) Since, BC || MD and CD is transversal.

\(\therefore \) \(\angle{ADM}\) = \(\angle{ACB}\)

(Corresponding angles)

But, \(\angle{C}\) = \(90^\circ\)

\(\therefore \) \(\angle{ADM}\) = \(90^\circ\)

Hence, proved that MD is perpendicular to AC.

iii) Now, in \(\triangle{ADM}\) and \(\triangle{ACM}\), we have,

AD = CD ...(from (i))

DM = MD ...(Common side)

\(\angle{ADM}\) = \(\angle{MDC}\) = \(90^\circ\) ...(Proved)

\(\therefore \) \(\triangle{ADM}\) \(\displaystyle \cong \) \(\triangle{CDM}\) ...(By SAS rule)

\(\therefore\) CM = AM ...(ii)(By CPCT)

Also, M is the mid point of AB.

\(\Rightarrow \) AM = BM = (\(\frac{1}{2} \) ) AB ...(iii)

Thus, from (ii) and (iii),

CM = AM = (\(\frac{1}{2} \) ) AB

Hence, proved.

- ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see figure). AC is a diagonal. Show thati) SR || AC and SR = (1/2) AC ii) PQ = SRiii) PQRS is a parallelogram.
- ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively. Show that the quadrilateral PQRS is a rectangle.
- ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively. Show that the quadrilateral PQRS is a rhombus.
- ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see figure). Show that F is the mid-point of BC.
- In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see figure). Show that the line segments AF and EC trisect the diagonal BD.
- Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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