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About Constructions — Class 10 ICSE

Construct tangents to circles, circumscribed and inscribed circles of triangles. This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 17). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Constructions

  • Introduction to Geometrical Constructions & Basics
  • Constructing Tangents to a Circle (Part 1: From a point on the circle)
  • Constructing Tangents to a Circle (Part 2: From an external point)
  • Constructing Inscribed and Circumscribed Circles of a Triangle
  • Summary, Verification, and Exam Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Constructions — solved examples for Class 10 ICSE

Example 1easy

To construct a tangent to a given circle at a point P on the circle, which of the following is the essential first step after drawing the radius OP?
  1. A)A. Draw another radius not passing through P.
  2. B)B. Draw a line parallel to OP.
  3. C)C. Draw a line perpendicular to OP at P.
  4. D)D. Find the midpoint of OP.

Step-by-step solution

  1. A key property of tangents is that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
  2. Therefore, to construct a tangent at point P, one must first draw the radius OP, and then construct a line perpendicular to OP passing through P.

Answer: C. Draw a line perpendicular to OP at P.

Example 2medium

How many tangents can be drawn to a circle from a point lying outside the circle?
  1. A)A) Infinite
  2. B)B) One
  3. C)C) Two
  4. D)D) Three

Step-by-step solution

  1. A point located outside a circle allows for the construction of exactly two distinct tangents to that circle.
  2. These tangents connect the external point to two different points on the circle, each touching the circle at only one point.

Answer: C) Two

Example 3hard

A point P is 13 cm away from the center O of a circle with radius 5 cm. Tangents PA and PB are drawn from P to the circle. What is the area of the quadrilateral OAPB?
  1. A)30 cm²
  2. B)60 cm²
  3. C)65 cm²
  4. D)120 cm²

Step-by-step solution

  1. Since OA is the radius and PA is a tangent, ∠OAP = 90°. Similarly, ∠OBP = 90°.
  2. In right-angled ΔOAP, by Pythagoras theorem, OP² = OA² + PA². So, 13² = 5² + PA² => 169 = 25 + PA² => PA² = 144 => PA = 12 cm.
  3. The quadrilateral OAPB can be divided into two congruent right-angled triangles, ΔOAP and ΔOBP.
  4. Area of ΔOAP = (1/2) × base × height = (1/2) × OA × PA = (1/2) × 5 cm × 12 cm = 30 cm². Therefore, Area of OAPB = 2 × Area of ΔOAP = 2 × 30 cm² = 60 cm².

Answer: 60 cm²

Practice questions on Constructions

  1. Q1.easy

    When constructing tangents from an external point P to a circle with center O, an auxiliary circle is drawn with OP as diameter. The intersection points of this auxiliary circle with the original circle are the points of tangency. This method works because:
    1. A)A. The auxiliary circle is always concentric with the original circle.
    2. B)B. This is a standard construction method, but there's no specific geometric reason.
    3. C)C. The angle subtended by a diameter at any point on the circumference is 90°, and a tangent is perpendicular to the radius at the point of contact.
    4. D)D. The diameter of a circle is always a tangent to another circle.
    Show answer

    Answer: C. The angle subtended by a diameter at any point on the circumference is 90°, and a tangent is perpendicular to the radius at the point of contact.

    Hint: Consider the angle formed when a point on a circle is connected to the ends of its diameter. How does this relate to the tangent-radius property?

  2. Q2.easy

    Which of the following geometric loci is used to locate the circumcenter of a triangle?
    1. A)A. The intersection point of two angle bisectors.
    2. B)B. The intersection point of two medians.
    3. C)C. The intersection point of two perpendicular bisectors of the sides.
    4. D)D. The intersection point of two altitudes.
    Show answer

    Answer: C. The intersection point of two perpendicular bisectors of the sides.

    Hint: The circumcenter is equidistant from the vertices of the triangle. Which lines represent points equidistant from two given points?

  3. Q3.easy

    To construct the incircle of a triangle, which specific lines inside the triangle must be drawn to find the center of the incircle?
    1. A)A. Perpendicular bisectors of the sides
    2. B)B. Altitudes
    3. C)C. Medians
    4. D)D. Angle bisectors
    Show answer

    Answer: D. Angle bisectors

    Hint: The incenter is equidistant from the sides of the triangle. Which lines represent points equidistant from two given lines?

  4. Q4.medium

    Which of the following is the **first** crucial step when constructing tangents to a circle from an external point P?
    1. A)A) Draw a line segment from P to the center O.
    2. B)B) Bisect the line segment PO.
    3. C)C) Draw a circle with P as center.
    4. D)D) Locate the points of intersection on the original circle.
    Show answer

    Answer: A) Draw a line segment from P to the center O.

    Hint: To find the midpoint for the construction of tangents, you first need a segment to bisect.

  5. Q5.medium

    If PA and PB are two tangents drawn to a circle with center O from an external point P, then what is the relationship between the lengths of PA and PB?
    1. A)A) PA = 2 × PB
    2. B)B) PA > PB
    3. C)C) PA < PB
    4. D)D) PA = PB
    Show answer

    Answer: D) PA = PB

    Hint: A fundamental theorem in circle geometry states a property about the lengths of tangents drawn from an external point.

  6. Q6.medium

    For an obtuse-angled triangle, where does its circumcenter lie?
    1. A)A) At one of the vertices
    2. B)B) Inside the triangle
    3. C)C) On one of the sides
    4. D)D) Outside the triangle
    Show answer

    Answer: D) Outside the triangle

    Hint: Visualize an obtuse triangle and consider how its perpendicular bisectors might intersect.

  7. Q7.hard

    In ΔABC, the angle bisectors of ∠B and ∠C meet at I (the incenter). If ∠BIC = 125°, what is the measure of ∠BAC?
    1. A)50°
    2. B)60°
    3. C)70°
    4. D)80°
    Show answer

    Answer: 70°

    Hint: Remember the relationship between the angle at the incenter and the opposite vertex angle of the triangle.

  8. Q8.hard

    A student is asked to construct the circumcircle of an obtuse-angled triangle. Which statement about the circumcenter (O) of an obtuse-angled triangle is always true?
    1. A)The circumcenter lies inside the triangle.
    2. B)The circumcenter lies on one of the sides of the triangle.
    3. C)The circumcenter lies outside the triangle.
    4. D)The circumcenter coincides with one of the vertices.
    Show answer

    Answer: The circumcenter lies outside the triangle.

    Hint: Consider the positions of the circumcenter for acute, right, and obtuse triangles. The perpendicular bisectors define its location.

  9. Q9.hard

    To construct a tangent to a circle at point P on the circle, without using the center, Ravi followed these steps:
    1. Draw a chord PQ.
    2. Take a point R on the major arc PQ.
    3. Join PR and QR.
    4. Construct ∠XPR equal to ∠PQR.
    Which geometrical theorem is Ravi relying on to ensure that the line XP is indeed the tangent?
    1. A)Angle in a semicircle theorem
    2. B)Alternate Segment Theorem
    3. C)Tangent-Secant Theorem
    4. D)Angles subtended by the same arc theorem
    Show answer

    Answer: Alternate Segment Theorem

    Hint: Think about the relationship between the angle between a tangent and a chord through the point of contact, and angles in the alternate segment.

These are 9 of the 60 questions available for Constructions. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.