Constructions

Constructions is Chapter 11 of NCERT Class 10 Mathematics and a separate chapter category on Teachoo. It covers division of a line segment in a given ratio, construction of triangles similar to a given triangle for scale factors greater than or less than one, and construction of tangents to a circle. Teachoo explains both the compass-and-straightedge procedure and the geometric justification behind every step.

Dividing a line segment in a ratio

To divide segment AB in ratio m, an auxiliary ray is drawn from A. Equal steps are marked along the ray, and the final mark is joined to B. A parallel through the appropriate intermediate mark intersects AB at the required point. The Basic Proportionality Theorem guarantees the specified ratio.

The construction is exact because it creates proportional segments through parallel lines. Students should preserve equal compass widths and construction lines rather than estimating the division with a ruler.

Constructing similar triangles

Given a triangle, a similar triangle can be constructed with scale factor greater than one for enlargement or less than one for reduction. Equal divisions on an auxiliary ray and parallel-line construction produce proportional corresponding sides. The AA condition establishes similarity.

The numerator and denominator of the scale factor determine how many equal steps are marked and which step is connected. A rough expectation of the final triangle’s size helps detect reversal.

Constructing tangents

For a tangent from an external point P to a circle with centre O, join OP and construct its midpoint. A circle with that midpoint as centre and radius equal to half OP intersects the original circle at tangent points. The angle in the semicircle is 90°, so each radius to a contact point is perpendicular to the constructed line.

Teachoo also covers constructing a tangent when the original circle’s centre is not given, which first requires locating the centre through perpendicular bisectors of chords.

Topics available on Teachoo

• dividing a line segment in a given ratio;

• constructing similar triangles for scale factor greater than one;

• constructing similar triangles for scale factor less than one;

• tangent from an external point;

• tangent when the circle’s centre is not given;

• NCERT examples and construction justifications.

Learning outcomes

Students should be able to perform each construction accurately, label every point and justify it using proportionality, similarity or circle theorems. They should distinguish enlargement from reduction and explain why a constructed tangent is perpendicular to the corresponding radius.

How Teachoo helps

Teachoo breaks each construction into visible steps. Perform the construction while reading, then repeat it without guidance. Write a short proof after the procedure. Board marks are often divided between the diagram, construction steps and justification, so all three must be complete.

Important concept connections

Every construction applies an earlier theorem. Segment division uses the Basic Proportionality Theorem, similar-triangle construction uses proportional sides and AA similarity, and tangent construction uses the right angle in a semicircle plus the radius–tangent theorem. This makes Constructions a practical synthesis chapter. If the underlying theorem is weak, the procedure becomes fragile memorisation; if the theorem is clear, the steps can be reconstructed.

Board-exam and competency preparation

Construction marks depend on evidence. Keep every relevant arc, auxiliary ray, equal subdivision and parallel line visible. Write steps in the order performed and use consistent point names between the text and figure. The justification should cite the theorem that transfers a ratio, establishes similarity or proves perpendicularity.

Before beginning, predict whether the constructed triangle should be larger or smaller and where the division point should lie. This catches inverted scale factors and ratios. In tangent construction, verify that the source point is outside the circle and that the contact radius is perpendicular. If the centre is missing, show the perpendicular-bisector construction that locates it; do not estimate it visually.

Quick revision checklist

Complete all five Teachoo construction types without copying, label and write their steps, and add a two- or three-line justification for each. Inspect the final diagram for retained arcs, parallel marks and exact contact points.

Common mistakes to avoid

Do not change compass width when marking equal divisions. Retain auxiliary rays, parallels and arcs. Use a sharp pencil and label the final required figure clearly. A visually plausible tangent or ratio is not enough; the construction must guarantee it.

Deeper reasoning and concept connections

In Constructions, fluency means more than repeating a procedure. Students should be able to recognise the underlying structure when the numbers, diagram, wording or orientation changes. A useful routine is: identify the mathematical objects, list the known and unknown quantities, state the governing property, carry out the steps and verify that every condition has been used.

Look for connections within the chapter as well. A definition usually leads to a representation; the representation reveals a pattern; and the pattern supports a rule or calculation. Explaining this chain improves retention and helps with case-based questions. It also prevents the common mistake of selecting a formula simply because its symbols resemble the numbers in the question.

How to solve unfamiliar and competency-based questions

Use a five-step response: interpret, represent, select, solve and verify. Interpret the wording; represent the information; select a definition, property or formula; solve without skipping the logical step; and verify through substitution, estimation, measurement or an alternative representation. This routine works for direct exercises as well as case-based questions.

If information appears unnecessary, ask whether it establishes a hidden condition. If information is missing, state what cannot be determined instead of inventing a value. In written answers, name the rule being used. Clear reasoning helps a teacher award method marks and also makes the page easier for a student—or an AI answer system—to retrieve for the precise doubt being asked.

What complete mastery looks like

For Constructions, a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.

Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.

Additional frequently asked questions

What should a student know before starting Constructions?

Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.

How can a student check an answer in Constructions?

Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.

How many questions are enough for strong preparation?

There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.

How should Teachoo solutions be used without becoming dependent on them?

Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.

Frequently asked questions

Why are parallel lines used to divide a segment?

The Basic Proportionality Theorem transfers the chosen ratio from equally spaced auxiliary marks to the original segment.

How do I know whether the similar triangle should be larger?

A scale factor greater than one enlarges it; a factor between zero and one reduces it.

How is a tangent justified?

The construction creates a right angle between the radius and the line at the contact point.

Do not copy the diagram mechanically. Know which theorem makes every construction step valid.