NCERT Solutions Class 9 Maths Chapter 12 Quadrilaterals on Padhaiwise

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 12 Quadrilaterals Exercise Set 12.1

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 12 — Exercise Set 12.1#

Definition (from the book): for four distinct points $A, B, C, D$ in a plane, with no three collinear, the points on segments $AB$, $BC$, $CD$, $DA$ form quadrilateral ABCD if every such point other than $A, B, C, D$ lies on exactly one of these segments. A quadrilateral is convex when all its internal angles are less than 180° — equivalently, when its diagonals intersect.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 12 Quadrilaterals Exercise Set 12.2

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 12 — Exercise Set 12.2#

Theorem 1: In a parallelogram, (a) opposite sides are equal, (b) opposite angles are equal, (c) the diagonals bisect each other.

Theorems 2–5 (tests for a parallelogram): a quadrilateral is a parallelogram if its opposite sides are equal, or its opposite angles are equal, or its diagonals bisect each other, or one pair of opposite sides is both equal and parallel.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 12 Quadrilaterals Exercise Set 12.3

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 12 — Exercise Set 12.3#

Theorem 6 (Midpoint Theorem): the segment joining the midpoints of two sides of a triangle is parallel to the third side and has half its length.

Theorem 7 (Converse): the line through the midpoint of one side, parallel to another side, bisects the third side.

Theorem 9 (Varignon): the midpoints of the four sides of a quadrilateral are the vertices of a parallelogram.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 12 Quadrilaterals Exercise Set 12.4

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 12 — Exercise Set 12.4#

In any tiling, the angles of the tiles that meet around a point must add up to exactly 360° — no gaps and no overlaps.

1. Justify why the plane cannot be tiled with a regular pentagon. #

Solution

Each angle of a regular pentagon is:

$$\dfrac{(5 - 2) \times 180°}{5} = 108°$$

Look at a corner of a tile. Around that point the angles must total 360°.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 12 Quadrilaterals End-of-Chapter Exercises

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 12 — End-of-Chapter Exercises#

Parallelogram tests: opposite sides equal; opposite angles equal; diagonals bisect each other; one pair of sides equal and parallel.

Midpoint Theorem and its converse (Theorem 7); Centroid Theorem (medians meet at a point dividing each in the ratio 2 : 1); Varignon’s theorem.

1. Using a fact about parallelograms, show how to tile the plane using any given triangle. (Hint: Can you use the parallelogram tiling in the introduction?) #

Solution

Take the triangle $ABC$ and a copy turned through 180° about the midpoint $M$ of side $BC$. The half-turn sends $B$ to $C$ and $C$ to $B$, and sends $A$ to a new point $D$ with $M$ the midpoint of $AD$.