Chapter Overview
Chapter 12 introduces quadrilaterals and their properties, proves tests for parallelograms, develops the midpoint theorem and studies how shapes can tile a plane.
These worked solutions show the mathematical reasoning behind selected exercise questions and end-of-chapter problems.
Textbook-check note: Use the exact question numbers and diagrams in your copy of Ganita Manjari. Where data are read from a graph, answers are marked as estimates. This page does not reproduce every printed exercise verbatim.
Important Concepts
| Concept | Explanation |
|---|---|
| Quadrilateral | A four-sided polygon; the sum of interior angles of a simple quadrilateral is 360°. |
| Parallelogram | A quadrilateral whose two pairs of opposite sides are parallel. |
| Parallelogram tests | A quadrilateral is a parallelogram if both pairs of opposite sides are equal, both pairs of opposite angles are equal, or its diagonals bisect each other. |
| Midpoint theorem | The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. |
| Tiling | Covering a plane without gaps or overlaps using copies of a shape. |
Exercise Solutions — Chapter 12
Exercise Set 12.1 — Understanding quadrilaterals
Question 1 — Sides and angles of ABCD
(i) Sides adjacent to AB are BC and AD; the side opposite AB is CD.
(ii) Angles adjacent to ∠A are ∠B and ∠D; the angle opposite ∠A is ∠C.
(iii) Adjacent elements share an endpoint or side, while opposite sides and opposite angles are across the quadrilateral from one another.
Exercise Set 12.2 — Parallelograms
Question 1 — True or false
(i) A parallelogram with a right angle is a rectangle — True. Adjacent angles are supplementary, so if one is 90°, all four are 90°.
(ii) A rhombus with perpendicular diagonals is a square — False. The diagonals of every rhombus are perpendicular, but a rhombus need not have four right angles.
(iii) A parallelogram with equal diagonals is a rectangle — True.
Question 2 — A diagonal bisects an angle
In parallelogram ABCD, suppose diagonal AC bisects ∠A, so ∠BAC = ∠DAC. Because AB ∥ DC and AD ∥ BC, alternate interior angles give ∠BAC = ∠DCA and ∠DAC = ∠BCA. Hence ∠DCA = ∠BCA, so AC also bisects ∠C. In triangle ADC, ∠DAC = ∠DCA, so AD = DC. Opposite sides of a parallelogram are equal; therefore all four sides are equal and ABCD is a rhombus.
Question 3 — Converses of diagonal properties
(i) No. A quadrilateral with angle-bisecting diagonals need not be a rhombus unless the other required conditions are also known. An orthodiagonal kite can have one diagonal bisecting the other without being a rhombus. Check the exact wording and diagram in the printed exercise before using this result for marking.
(ii) Yes. If the diagonals bisect each other, the quadrilateral is a parallelogram. If they also meet at right angles, adjacent sides are equal (the right triangles formed by the diagonals have equal half-diagonals and a common included structure), so the parallelogram is a rhombus.
(iii) No. Equal diagonals alone do not guarantee a rectangle; an isosceles trapezium is a counterexample. Add the condition that the quadrilateral is a parallelogram, and equal diagonals then imply a rectangle.
Exercise Set 12.3 — Midpoints and medians
Question 1 — Midpoint construction in a triangle
Let P, Q and R be midpoints of AB, AC and BC in triangle ABC. By the midpoint theorem, PQ = BC/2, PR = AC/2, and QR = AB/2. The four small triangles have corresponding side lengths AB/2, BC/2 and AC/2; therefore they are congruent by SSS.
Exercise Set 12.4 — Tiling
Question 1 — Why a regular pentagon cannot tile the plane
Each interior angle is ((5 − 2)×180°)/5 = 108°. Around a point, 3 such angles total 324° (a gap of 36°), while 4 total 432° (an overlap). Since neither gives 360°, regular pentagons cannot tile the plane on their own.
End-of-Chapter Exercises
Question 1 — Tiling with triangles
Two congruent copies of any triangle can be placed with one rotated 180° about the midpoint of a common side. They form a parallelogram, which tiles the plane. Hence copies of the original triangle also tile the plane.
Question 2 — Tiling with a non-convex quadrilateral
A non-convex quadrilateral’s interior angles still sum to 360°. Copies can be arranged around each vertex using the same side-to-side rotation construction; rotating a copy through 180° about the midpoint of each shared side creates matching edges and continues the tiling without gaps or overlaps.
One Minute Revision
- Angles of a simple quadrilateral add up to 360°.
- Opposite sides and opposite angles of a parallelogram are equal.
- In a parallelogram, diagonals bisect each other.
- The midpoint segment in a triangle is parallel to the third side and half its length.
- A regular pentagon’s interior angle is 108°, which cannot fit around a point an integer number of times to make 360°.
Frequently Asked Questions
1. What is the sum of a quadrilateral’s interior angles?
360° for a simple quadrilateral, whether convex or non-convex.
2. What does the midpoint theorem say?
The segment joining midpoints of two sides of a triangle is parallel to the third side and half its length.
3. Can a regular pentagon tile the plane by itself?
No. Each interior angle is 108°, and whole multiples of 108° cannot total 360° around a point.