NCERT Class 9 Maths Solutions: Quadrilaterals

Question:

Properties of parallelograms (opposite sides equal and parallel, diagonals bisect each other), properties of a rhombus (all sides equal), properties of a rectangle (all angles right angles), Pythagorean theorem, congruence of triangles.


Let ABCD be a square.
Step 1: Prove diagonals are equal.
Consider triangles ABC and BAD.
AB = BA (common side)
BC = AD (sides of a square are equal)
Angle ABC = Angle BAD = 90 degrees (angles of a square are right angles)
By SAS congruence rule, triangle ABC is congruent to triangle BAD.
Therefore, AC = BD (corresponding parts of congruent triangles are equal).
Hence, the diagonals of a square are equal.

Step 2: Prove diagonals bisect each other.
Consider triangles AOB and COD (where O is the intersection of diagonals AC and BD).
AB = CD (sides of a square are equal)
Angle OAB = Angle OCD (alternate interior angles as AB || DC)
Angle OBA = Angle ODC (alternate interior angles as AB || DC)
By ASA congruence rule, triangle AOB is congruent to triangle COD.
Therefore, AO = CO and BO = DO (corresponding parts of congruent triangles are equal).
This shows that the diagonals bisect each other.

Step 3: Prove diagonals bisect each other at right angles.
Consider triangles AOB and AOD.
AB = AD (sides of a square are equal)
AO = AO (common side)
BO = DO (proved in Step 2, diagonals bisect each other)
By SSS congruence rule, triangle AOB is congruent to triangle AOD.
Therefore, Angle AOB = Angle AOD (corresponding parts of congruent triangles are equal).
Since Angle AOB and Angle AOD form a linear pair, Angle AOB + Angle AOD = 180 degrees.
So, Angle AOB + Angle AOB = 180 degrees.
2 * Angle AOB = 180 degrees.
Angle AOB = 90 degrees.
Hence, the diagonals bisect each other at right angles.

Question:

Midpoint Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side.
Properties of a Rectangle: Opposite sides are equal and parallel. All angles are 90 degrees.
Properties of a Rhombus: All four sides are equal in length.


Step 1: Visualize the rectangle ABCD and the midpoints P, Q, R, S on its sides.
Step 2: Consider triangle ABC. P is the midpoint of AB and Q is the midpoint of BC. Apply the Midpoint Theorem to triangle ABC. What can you say about the line segment PQ?
Step 3: Similarly, consider triangle ADC. S is the midpoint of DA and R is the midpoint of CD. Apply the Midpoint Theorem to triangle ADC. What can you say about the line segment SR?
Step 4: Now, consider triangle BCD. Q is the midpoint of BC and R is the midpoint of CD. Apply the Midpoint Theorem to triangle BCD. What can you say about the line segment QR?
Step 5: Also, consider triangle ABD. P is the midpoint of AB and S is the midpoint of DA. Apply the Midpoint Theorem to triangle ABD. What can you say about the line segment PS?
Step 6: From the applications of the Midpoint Theorem in the previous steps, you will have relationships between the sides PQ, QR, RS, and SP with the diagonals of the rectangle AC and BD.
Step 7: Use the properties of a rectangle, specifically that its diagonals are equal in length (AC = BD).
Step 8: By comparing the lengths of PQ, QR, RS, and SP derived from the Midpoint Theorem and the property of equal diagonals, show that all four sides of the quadrilateral PQRS are equal.
Step 9: Conclude that since all four sides of PQRS are equal, it is a rhombus.