Geometry · Mathematics

Geometry Construction Lab: Compass & Ruler Simulations

Eighteen compass-and-straightedge constructions, each built one step at a time with the reason for every step: perpendicular and angle bisectors, the standard angles, √5 and any √n on the number line, the parts of a circle, chords and the centre, equal chords, the circle through three points, tangents at and from a point, circles through two points or touching a line, angles in a segment, cyclic quadrilaterals, the incircle and circumcircle, intersecting chords, secants and tangents, and √x by the semicircle. A live panel checks every theorem on the construction as you drag it.

Interactive virtual lab · no download · works on phones and laptops

Chapter: Geometry & Constructions
For: Class 7–12 · JEE
Time: ~35 min

Geometry Construction Lab is a Pro simulation

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How to use it

1Pick a module (grouped by Class 7–8, 9–10 and 11–12 · JEE) and read what is given.
2Press ‘Build it step by step’ or step through; each step shows why it works. Drag the orange points or type values.
3Watch the live check, then read the theorem and proof, the common mistakes (each can be shown on the board), the challenges, and practise.

What to notice

A construction is exact: the compass carries lengths and the arcs find points, so nothing is measured or rounded.
Arcs about A and B cross only when the radius is more than half of AB — at exactly half they touch once, at the midpoint.
√5 is the hypotenuse of legs 2 and 1; its point on the number line is exact even though 2.236… never ends.
The perpendicular bisectors of chords all pass through the centre; angle bisectors meet at the incentre.
Through a point P, every chord or secant gives the same product PA·PB — r² − OP² inside, PT² = OP² − r² outside.

For teachers

A ready-made teaching kit comes with this lab

Lesson plan
35 min, 4 steps
Ask the class
5 questions with answers
Quick quiz
7 questions, printable
Key formulas
6 formulas

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About the Geometry Construction Lab simulation

A geometric construction uses only an unmarked straightedge and a compass. The compass carries a length from one place to another and the arcs it draws find new points where they cross, so a construction is exact — nothing is measured and nothing is rounded. This lab builds each construction in front of you one step at a time, and every step comes with the reason it works. A live panel recomputes the theorem from the constructed points as you drag them: PA = PB on the perpendicular bisector, the two halves of a bisected angle, OB² = 5 for the hypotenuse of legs 2 and 1, the equal tangents from an external point.

The eighteen modules run from Class 7–8 (perpendicular and angle bisectors, 60°, 30°, 15°, 120°, 90°, 45° and 75° without a protractor, √5 and any √n on the number line, the parts of a circle) through the Class 9–10 circle theorems (the perpendicular from the centre bisects a chord, equal chords, the circle through three points, tangents at and from a point, circles through two points or touching a line, angles in a segment and in a semicircle, cyclic quadrilaterals, the incircle and circumcircle) to a JEE extension on the power of a point and on constructing √x for any length. Each module lists what is given and the tools allowed, states the theorem with a proof, shows the common mistakes on the board, and ends with challenges and randomised practice.

Board syllabi differ here: the rationalised NCERT books (2023) dropped the separate Constructions chapters from Classes 9 and 10, while many state boards and ICSE still set them. The circle theorems and locating √2, √3 and √5 on the number line remain in the NCERT Class 9 and 10 books.

Also covers: geometric construction, compass and straightedge, compass and ruler, perpendicular bisector, angle bisector, construct 60 degree angle, construct 90 degree angle, construct 45 degree angle, construct 30 degree angle, construct 15 degree angle, root 5 on number line, square root spiral, construct root 2, circle theorems, chord, secant, tangent, perpendicular from centre bisects chord, equal chords, circle through three points, circumcircle, incircle, tangent at a point, tangents from external point, angle in a semicircle, angles in the same segment, cyclic quadrilateral, intersecting chords theorem, secant tangent theorem, power of a point, geometric mean construction.

Frequently asked questions

Why must the compass radius be more than half the segment for the perpendicular bisector?

The arcs about A and B meet where a point is r from both. The closest any point can be to both is AB/2, at the midpoint. So with r > AB/2 the arcs cross twice and give the line; at exactly AB/2 they touch once, at the midpoint, which is not enough to draw a line; below that they never meet.

How do you construct √5 on the number line?

Mark O at 0 and A at 2. Construct a perpendicular at A and mark B one unit up. By Pythagoras OB² = 2² + 1² = 5, so OB = √5. Open the compass to OB, put the needle at O and mark where the arc meets the number line: that point is exactly √5, though its decimal 2.236… never ends.

How do you construct tangents to a circle from a point outside it?

Join the centre O to the point P and bisect OP at M. Draw the circle with centre M through O and P; it cuts the given circle at T₁ and T₂. PT₁ and PT₂ are the tangents, because the angle OT₁P is in a semicircle and so is 90°. Both have length √(OP² − r²).

How do you find the centre of a circle using constructions?

Draw two chords and construct the perpendicular bisector of each. Every perpendicular bisector of a chord passes through the centre, so the two meet at it. The same method gives the circle through any three points that are not on one line.

What is the intersecting chords theorem?

If two chords AB and CD meet at P inside a circle, then PA·PB = PC·PD. Joining AC and BD gives similar triangles (equal angles in the same segment and vertically opposite angles). The common product is r² − OP²; outside the circle the same idea gives PT² = PA·PB for a tangent and a secant.