Constructions Class 10 Maths Notes (Chapter 11)

Must-Remember Key Points

• {"point":"Only an ungraduated ruler and a compass are permitted for geometric constructions."}
• {"point":"Always show your construction arcs. Do not erase them, as they carry marks and show your method."}
• {"point":"Justification of the construction is as important as the construction itself and is often asked in exams."}
• {"point":"To divide a line segment in the ratio m:n, you need to mark a total of (m+n) equal arcs on a ray drawn at an acute angle."}
• {"point":"The justification for dividing a line segment relies on the Basic Proportionality Theorem (Thales' Theorem)."}
• {"point":"The justification for constructing tangents relies on the property that the angle in a semicircle is 90° and a tangent is perpendicular to the radius at the point of contact."}
• {"point":"The lengths of the two tangents drawn from an external point to a circle are always equal."}
• {"point":"No tangent can be drawn to a circle from a point lying inside it."}

Key Terms for Constructions

Line Segment

A part of a line that is bounded by two distinct end points.

Ratio

A comparison of two quantities, showing their relative sizes. In constructions, used as m:n to divide a line segment.

Tangent

A line that touches a circle at exactly one point, called the point of contact.

Secant

A line that intersects a circle at two distinct points.

Point of Contact

The single point where a tangent line touches a circle.

Perpendicular Bisector

A line that divides another line segment into two equal parts at a 90° angle.

Justification

The mathematical proof or reasoning that demonstrates why a given construction method is valid and accurate.

Process 1: Dividing a Line Segment in a Given Ratio (m:n)

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Concept Explained: Justification for Tangent Construction

Why does the method for constructing tangents from an external point work? The entire process is built on a fundamental theorem from the Circles chapter. Let's say we have a circle with center O and an external point P. We want to construct tangents from P to the circle.

The steps involve joining OP and then constructing its perpendicular bisector to find the midpoint, M. With M as the center and MO (or MP) as the radius, we draw a new circle. This new circle intersects our original circle at two points, say T₁ and T₂. We then join PT₁ and PT₂ to get our tangents.

The justification lies here: Consider the point T₁. It lies on the new circle with diameter OP. The angle subtended by a diameter at any point on the circumference of a circle is a right angle (90°). Therefore, ∠OT₁P = 90°. This means that the line OT₁ is perpendicular to the line PT₁. Since OT₁ is the radius of the original circle and PT₁ is perpendicular to it at point T₁, PT₁ must be a tangent to the circle at T₁. The same logic applies to prove that PT₂ is also a tangent. This elegant proof combines properties of perpendicular lines, circles, and tangents, making it a crucial concept to understand.

Process 2: Constructing Tangents to a Circle from an External Point

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Exam Tips for Constructions

Examiners look for precision and clear methodology. The most common mistake students make is erasing the construction arcs after drawing the final figure. Never erase your arcs! They are proof of your method and carry marks. Use a sharp, light H or 2H pencil for construction lines and a darker HB pencil for the final required lines (like the tangents or the divided segment) to make your answer sheet clean and easy to evaluate. Also, ensure your labeling (A, B, O, P, T₁, etc.) is neat and legible.

Practice Questions with Solutions

• To divide a line segment in the ratio 4:7, how many total points must be marked on the ray? You must mark 4 + 7 = 11 points.
• What is the angle between a tangent and the radius at the point of contact? The angle is always 90 degrees.
• How many tangents can you draw to a circle from a point on the circle? Exactly one.
• What is the first step in constructing a tangent from an external point P to a circle with center O? Join the center O to the external point P to form the line segment OP.

Frequently Asked Questions (FAQs)

Frequently Asked Questions