Question

If the diagonals of a quadrilateral intersect at right angles then the figure obtained by joining the mid points of the adjacent sides of the quadrilateral is a rectangle. State whether the statement is true or false.

Answer 1

Hint: Now we have a quadrilateral whose diagonal intersect at right angle. Now with diagonals as base, consider each possible triangle and use the mid-point theorem. Hence we will get conditions to prove that the quadrilateral formed by joining midpoints is parallelogram. Now we know that angles made by lines are equal to angles made by parallels hence we can prove that any one of the angles of the quadrilateral obtained is the right angle. And we know that if a parallelogram has one angle as ${{90}^{\circ }}$ then the parallelogram is a rectangle.

Complete step-by-step answer:
Now first let us consider a figure with the following conditions.

Here ABCD is any quadrilateral with AC perpendicular to BD.
And P, Q, R, S are midpoints of AB, BC, CD and DA respectively.
Now first consider triangle ABC.
Now according to Mid-point theorem we have the line joining midpoint of two sides is parallel to third side
Hence using this theorem in triangle ABC we have PQ parallel to AC since P and Q are mid points of AB and BC respectively
Hence we have $PQ\parallel AC................(1)$
Now consider triangle ADC.
Now again according to Mid-point theorem we have the line joining midpoint of two sides is parallel to third side
Hence using this theorem in triangle ADC we have SR parallel to AC since S and R are mid points of AD and DC respectively.
Hence we have $SR\parallel AC..........(2)$
Hence from equation (1) and equation (2) we get
$SR\parallel PQ........................(3)$
Similarly Now consider triangle ABD.
We can say by mid-point theorem $PS\parallel BD.......................(4)$
And In triangle CBD
We have again by mid point theorem that $QR\parallel BD......................(5)$
Hence from equation (4) and equation (5) we have
$PS\parallel QR.................(6)$
 Now from equation (3) and equation (6) we have
$PS\parallel QR$ and $SR\parallel PQ$
Hence PQRS is a parallelogram.
Now from equation (1) we have $PQ\parallel AC$
And from equation (5) we have $QR\parallel BD$
But we are given that AC is perpendicular to BD
And we know that angle between two lines is equal to angle between their parallels
Hence we can say that PQ is perpendicular to QR
Hence we have $\angle PQR={{90}^{\circ }}$
Now once we have one angle of parallelogram as ${{90}^{\circ }}$ we can say that the parallelogram is a rectangle.
Hence the statement is true.

Note: Now note that Parallelogram has opposite angles equal. Hence if one of the angles is ${{90}^{\circ }}$ then the angle opposite it also becomes ${{90}^{\circ }}$ . Now in a quadrilateral the sum of angles is ${{360}^{\circ }}$ and now we have two angles as ${{180}^{\circ }}$ hence the sum of rest two angles will be ${{360}^{\circ }}-{{180}^{\circ }}={{180}^{\circ }}$
Now also the rest two angles are opposite and equal and hence each angle is equal to $\dfrac{{{180}^{\circ }}}{2}={{90}^{\circ }}$