What is financial infrastructure?

The network of banks, payment systems and stock markets that helps manage money

What are the main things a bank does?

Holds deposits and lends money as loans

How do banks make money?

They pay lower interest on deposits and charge higher interest on loans

Earning interest on previous interest as well as on the original amount

Which points lie on the perpendicular bisector of XY?

Every point equidistant from X and Y.

How is an angle bisected with ruler and compass?

By constructing two congruent triangles (SSS), so the dividing ray makes equal angles.

How do you construct a 60° angle?

By constructing an equilateral triangle, whose angle is \(60^\circ\).

How can a 45° angle be made?

By constructing a \(90^\circ\) angle and bisecting it.

Why do six equilateral triangles form a regular hexagon?

Their six \(60^\circ\) angles add to \(360^\circ\) around the centre with no gap or overlap.

Covering a region with shapes leaving no gaps or overlaps.

Can a 5\u00d77 grid be tiled with 2\u00d71 tiles?

No — it has 35 unit squares, an odd number, but each tile covers two squares.

How does black-and-white colouring help in tiling?

A tileable region must have equal black and white squares; unequal counts prove it cannot be tiled.

Constructions and Tilings

Q: Which regular polygons can tile the plane?

Squares, equilateral triangles and regular hexagons.

Class 7 Mathematics · Ganita Prakash Part 2 · Chapter 6

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Five congruence conditions to equations

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HCF and LCM by prime factors, Ch 3
Multiplying and dividing decimals, Ch 4
Mean, median and outliers, Ch 5
Ruler-and-compass constructions, Ch 6

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CBSE Class 7 · Mathematics

8 chapters: lakhs and crores to fractions

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Summary

This chapter develops precise geometric constructions using only an unmarked ruler and a compass, then explores tilings of the plane. You learn to construct the perpendicular bisector of a segment (justified by congruent triangles, since any point equidistant from the two endpoints lies on it), a \(90^\circ\) angle at a point on a line, and to bisect any angle using the SSS congruence of two triangles. From these you build \(45^\circ\), \(60^\circ\), \(30^\circ\) and \(15^\circ\) angles, copy a given angle exactly, draw a line parallel to a given line, and construct a regular hexagon (made of six congruent equilateral triangles, since six \(60^\circ\) angles fill the \(360^\circ\) around a point). Ancient Shulba-Sutra rope methods are highlighted. In tiling, you ask which grids can be covered by \(2\times1\) tiles without gaps or overlaps, using a clever black-and-white colouring argument to prove some regions cannot be tiled, and you see how squares, equilateral triangles and regular hexagons can tile the entire plane.

Perpendicular bisector and 90° angleAngle bisection and copying anglesConstructing 60°, 45°, 30° angles and parallel linesConstructing a regular hexagonTiling grids and the plane

Key terms

Bisection

Dividing a line segment or angle into two identical parts.

Perpendicular bisector

A line that bisects a segment and is perpendicular to it.

Angle bisector

A ray that divides an angle into two equal angles.

Regular hexagon

A six-sided polygon with equal sides and equal angles, each \(120^\circ\).

Tiling

Covering a region using shapes without gaps or overlaps.

Compass and (unmarked) ruler

The two basic tools used for exact geometric constructions.

Extra questions & answers

Try one from this chapter

A perpendicular bisector of a segment is a line that:

Flashcards & interactive practice