1. Chapter at a Glance
Geometry is not only about measuring — it is also about constructing, that is, drawing shapes accurately using simple tools. With just a ruler (straight edge) and a compass you can draw circles and arcs, make perfect squares and rectangles, discover properties of their diagonals and find points that are at equal distances from two given points. These skills are used by architects, engineers, tailors and artists.
- Using a compass to draw circles, arcs and curved designs.
- The meaning of centre, radius and diameter.
- Constructing squares and rectangles with given sides.
- Properties of the diagonals of rectangles and squares.
- Points equidistant from two given points (the perpendicular bisector).
- Constructing a rectangle when a side and a diagonal are given.
2. Geometry Tools and Circles
- Ruler: draws straight lines and measures lengths.
- Compass: has a pointed metal leg and a pencil leg. Keeping the pointed leg fixed and turning the pencil leg draws a circle or part of one (an arc).
- Set squares and protractor: help with right angles and angle measures.
Parts of a circle
All points on a circle are at the same distance from a fixed point called the centre. That distance is the radius. A line segment through the centre joining two points of the circle is a diameter; it is twice the radius.
Curved designs
By drawing arcs of equal radius from points placed at equal gaps on a line, you can make wave patterns, flower patterns (six overlapping circles of the same radius make a "flower of life") and many rangoli-like designs. These show how a single simple tool can create beautiful shapes.
3. Squares and Rectangles
| Property | Rectangle | Square |
|---|---|---|
| Angles | All four are 90° | All four are 90° |
| Opposite sides | Equal | Equal |
| All four sides | Not necessarily equal | Equal |
| Diagonals | Equal, bisect each other | Equal, bisect each other at 90° |
Every square is a rectangle (it has four right angles), but not every rectangle is a square.
Construct a rectangle with sides 6 cm and 4 cm
- Draw a line segment AB = 6 cm.
- At A, draw a perpendicular (90° angle) using a set square or protractor.
- With the compass set to 4 cm and centre A, cut the perpendicular at D. So AD = 4 cm.
- Similarly draw a perpendicular at B and mark C with BC = 4 cm.
- Join D and C. ABCD is the required rectangle. Check: DC should measure 6 cm.
Construct a square of side 5 cm using a compass
- Draw PQ = 5 cm and a perpendicular at P; mark S with PS = 5 cm.
- With centre Q and radius 5 cm, draw an arc above Q.
- With centre S and radius 5 cm, draw an arc cutting the first arc at R.
- Join QR and SR. PQRS is a square — all sides are 5 cm, and the angles are right angles.
4. Exploring Diagonals
A diagonal joins two opposite corners. A rectangle ABCD has two diagonals, AC and BD. Measure them in your construction and you will find:
- The two diagonals of a rectangle are equal in length.
- They cut each other into two equal halves — they bisect each other. Their meeting point O is equally far from all four corners.
- In a square, the diagonals also meet at right angles.
5. Points Equidistant from Two Points
Given two points A and B, which points are at the same distance from both? Use the compass:
- With centre A and any radius more than half of AB, draw arcs above and below AB.
- With centre B and the same radius, draw arcs cutting the first arcs at P and Q.
- P and Q are each equally far from A and B (both were drawn with the same radius).
- Join PQ. Every point on line PQ is equidistant from A and B.
The line PQ cuts AB exactly in half and at a right angle. It is called the perpendicular bisector of AB. You will study it in detail in higher classes.
Two houses A and B are 8 cm apart on a map. A well must be dug at the same distance from both houses and on the road AB. Where should it be?
Solution: The point on AB that is equidistant from A and B is its midpoint — 4 cm from each house, where the perpendicular bisector meets AB.
6. Constructing a Rectangle from a Side and a Diagonal
Suppose a rectangle has side AB = 5 cm and diagonal AC = 13 cm.
- Draw AB = 5 cm and a perpendicular to AB at B.
- With centre A and radius 13 cm, draw an arc cutting the perpendicular at C.
- Measure BC — it comes out as 12 cm (because 5, 12 and 13 fit the right-angle rule you will meet later).
- Complete the rectangle by drawing AD perpendicular at A with AD = BC, and join DC.
In a correct rectangle, both diagonals measure the same (13 cm here) and opposite sides are equal. If they are not, redraw the right angles more carefully.
7. Master Formula Table
| Fact | Rule |
|---|---|
| Diameter and radius | d = 2r |
| Rectangle angles | each 90°, total 360° |
| Rectangle sides | opposite sides equal |
| Rectangle diagonals | equal and bisect each other |
| Square diagonals | equal, bisect each other at 90° |
| Equidistant points from A and B | lie on the perpendicular bisector of AB |
| Distance from O to corners of a rectangle | half the diagonal |
8. Master Summary Table
| Construction | Tools | Key check |
|---|---|---|
| Circle / arc | Compass | Same radius throughout |
| Rectangle (two sides) | Ruler, set square, compass | Opposite sides and diagonals equal |
| Square (one side) | Ruler, compass | All sides equal, diagonals equal |
| Equidistant points | Compass, ruler | Same radius from both points |
| Rectangle (side + diagonal) | Ruler, compass, set square | Other diagonal equal |
9. Mind Map
- Circle, arc
- Radius, diameter
- Curved designs
- Right angles
- Equal opposite sides
- Step-wise construction
- Equal
- Bisect each other
- ⊥ in squares
- Arcs from both points
- Perpendicular bisector
10. Tips and Tricks
- Keep the compass screw tight so the radius does not change while drawing.
- Use a sharp pencil — thick lines cause errors of 1–2 mm.
- Draw light construction arcs and keep them; they show your method.
- Always verify a rectangle by measuring both diagonals.
- Changing the compass radius between the two arcs of an equidistance construction.
- Confusing radius with diameter (diameter is twice as long).
- Assuming the diagonals of a rectangle meet at 90° — only in a square.
- Drawing the perpendicular "by eye" instead of using a set square or compass.
11. Important One-liners
- All points of a circle are at the same distance (radius) from the centre.
- Diameter = 2 × radius.
- A compass draws circles and arcs and copies lengths.
- All angles of a rectangle are right angles.
- Opposite sides of a rectangle are equal.
- The diagonals of a rectangle are equal and bisect each other.
- The diagonals of a square bisect each other at right angles.
- Every square is a rectangle, but not every rectangle is a square.
- Points equidistant from A and B lie on the perpendicular bisector of AB.
- The midpoint of AB is the point on AB equidistant from A and B.
12. Exam-pattern Questions with Detailed Answers
The radius of a circle is 4.5 cm. What is its diameter?
- 2.25 cm
- 9 cm
- 13.5 cm
- 4.5 cm
Show answer
Diameter = 2 × radius = 2 × 4.5 = 9 cm.
Answer: (B) 9 cm
State two properties of the diagonals of a rectangle.
Show answer
The diagonals of a rectangle are equal in length, and they bisect each other (their meeting point divides each into two equal parts).
Answer: Equal; bisect each other
Is every rectangle a square? Give a reason.
Show answer
No. A rectangle needs only opposite sides equal; a square needs all four sides equal. A 6 cm × 4 cm rectangle is not a square.
Answer: No
The diagonals of rectangle PQRS meet at O. If PR = 14 cm, find OQ.
Show answer
The diagonals are equal, so QS = 14 cm. They bisect each other, so OQ = 14 ÷ 2 = 7 cm.
Answer: 7 cm
Write the steps to construct a square of side 4 cm using a ruler and compass.
Show answer
1. Draw AB = 4 cm. 2. Draw a perpendicular at A and mark D with AD = 4 cm. 3. With centres B and D and radius 4 cm, draw arcs meeting at C. 4. Join BC and DC.
ABCD has all sides 4 cm and right angles, so it is the required square.
Answer: As above (verify both diagonals are equal)
A and B are 10 cm apart. Two arcs of radius 7 cm are drawn from A and from B. Why do the arcs meet? Would they meet if the radius were 4 cm?
Show answer
Arcs from A and B can meet only if the radius is more than half of AB (here more than 5 cm). 7 cm > 5 cm, so they meet.
With radius 4 cm, 4 + 4 = 8 cm < 10 cm, so the arcs cannot reach each other.
Answer: Yes for 7 cm; no for 4 cm
How many circles of radius 3 cm can pass through a given point P?
- One
- Two
- Four
- Infinitely many
Show answer
Any point that is 3 cm from P can be a centre, and there are infinitely many such points (they form a circle around P).
Answer: (D) Infinitely many
In a square ABCD, the diagonals meet at O. What is the measure of ∠AOB?
Show answer
In a square, the diagonals bisect each other at right angles, so ∠AOB = 90°.
Answer: 90°
A rectangle has one side 8 cm and its diagonal 10 cm. Describe how to construct it and find the other side by measuring.
Show answer
Draw AB = 8 cm; draw a perpendicular at B. With centre A, radius 10 cm, cut the perpendicular at C. Measure BC — it is 6 cm. Draw AD ⊥ AB with AD = 6 cm and join DC.
(6, 8, 10 is a right-angle triple: 6² + 8² = 36 + 64 = 100 = 10².)
Answer: Other side = 6 cm
Two towns A and B are 12 km apart. A school must be equally far from both towns and exactly 10 km from A. How many such places are there? Explain with arcs.
Show answer
Points equally far from A and B lie on the perpendicular bisector of AB. Points 10 km from A lie on a circle of radius 10 km around A.
Since 10 > 6 (half of AB), this circle cuts the perpendicular bisector at two points — one on each side of AB.
Answer: 2 places
The perimeter of a rectangle is 28 cm and one side is 9 cm. Find the other side and describe the construction.
Show answer
l + b = 28 ÷ 2 = 14, so the other side = 14 − 9 = 5 cm.
Construct AB = 9 cm, perpendiculars at A and B, mark AD = BC = 5 cm with the compass, and join DC.
Answer: 5 cm
Six circles of radius 2 cm are drawn with centres at the corners of a regular hexagon of side 2 cm. What is special about the centre of the hexagon?
Show answer
In a regular hexagon of side 2 cm, the distance from the centre to each corner is also 2 cm (it is made of six equilateral triangles).
So every circle passes through the centre of the hexagon — all six circles meet there, making a flower pattern.
Answer: All six circles pass through the centre
13. Quick Revision Summary
- A compass draws circles and arcs; radius is fixed; diameter = 2 × radius.
- Rectangles: four right angles, opposite sides equal, diagonals equal and bisecting each other.
- Squares: all sides equal; diagonals also meet at 90°.
- Construct rectangles and squares using perpendiculars and compass arcs; always verify with diagonals.
- Points equidistant from A and B lie on the perpendicular bisector of AB.