ICSE Class 10

General instructions: Answer all questions. Marks are shown in brackets [ ].

Objective

1. 1.
   Construct a triangle $ABC$, given $AB = 4$ cm, $BC = 6$ cm and $\angle ABC = 90^\circ$. What is the key step to construct the circle passing through points $A$, $B$, and $C$?

   1. A. Draw perpendicular bisectors of $AB$ and $BC$ intersecting at $O$.
   2. B. Draw angle bisectors of $\angle ABC$.
   3. C. Join $A$ to $C$ and measure $AC$.
   4. D. Draw a circle with radius $AB$.
   [1]
2. 2.
   Using ruler and compass, construct a $\Delta ABC$ where $AB = 3\mathrm{cm}$, $BC = 4\mathrm{cm}$ and $\angle ABC = 90^{\circ}$. What is the radius of the circle circumscribing $\Delta ABC$?

   1. A. $2$ cm
   2. B. $2.5$ cm
   3. C. $3$ cm
   4. D. $5$ cm
   [1]
3. 3.
   In the diagram showing the construction of an inscribed circle of triangle ABC, the point labelled I represents which of the following?

   1. A. The circumcentre, where the perpendicular bisectors of the sides meet
   2. B. The incentre, where the angle bisectors of the triangle meet
   3. C. The orthocentre, where the altitudes of the triangle meet
   4. D. The centroid, where the medians of the triangle meet
   [1]
4. 4.
   In the diagram showing the construction of the circumscribing circle of triangle ABC, which labelled point represents the circumcentre of the triangle?

   1. A. The point where two altitudes of the triangle intersect
   2. B. The point where the perpendicular bisectors of any two sides meet
   3. C. The midpoint of the longest side of the triangle
   4. D. The point where the angle bisectors of the triangle intersect
   [1]
5. 5.
   An equilateral triangle of side 4 cm is drawn. Which step is NOT part of constructing its circumcircle?

   1. A. Draw perpendicular bisectors of two sides
   2. B. Find the intersection of the bisectors as centre
   3. C. Draw arcs of radius 4 cm from all vertices
   4. D. Draw circle with radius equal to distance from centre to vertex
   [1]
6. 6.
   Draw a circle of radius 2.5 cm. Mark a point P at a distance of 6.5 cm from the centre O of the circle. Using ruler and compass, draw two tangents to the circle from P and measure the length of each tangent. What is the length of each tangent?

   1. A. 4.0 cm
   2. B. 5.0 cm
   3. C. 6.0 cm
   4. D. 7.0 cm
   [1]
7. 7.
   In the diagram below, a regular hexagon with side 5 cm is shown. Which labelled part represents the radius of the circumscribed circle around the hexagon?

   1. A. The perpendicular bisector of a side of the hexagon
   2. B. The line segment from the center to a vertex of the hexagon
   3. C. The side length of the hexagon (5 cm)
   4. D. The line joining the midpoints of two opposite sides
   [1]
8. 8.
   Construct a regular hexagon of side $3.5$ cm. To construct its circumcircle, which of the following is the correct method?

   1. A. Draw perpendicular bisectors of two adjacent sides to find the centre.
   2. B. Use the intersection of diagonals as the centre of the circumcircle.
   3. C. Construct the incircle first and use its centre for the circumcircle.
   4. D. Find the midpoint of one side and use it as the centre.
   [1]
9. 9.
   Use ruler and compass only to draw a circle of radius 4 cm with centre O. Mark a point P outside the circle at a distance of 7 cm from O. Construct two tangents to the circle from P and measure the length of any one tangent. What is the length of one tangent?

   1. A. 4.5 cm
   2. B. 5.7 cm
   3. C. 6.0 cm
   4. D. 6.5 cm
   [1]
10. 10.
    Construct a regular hexagon of side 5 cm and inscribe a circle in it. Which of the following steps correctly identifies the centre of the incircle?

    1. A. The intersection of the perpendicular bisectors of any two sides.
    2. B. The intersection of the angle bisectors of any two interior angles.
    3. C. The midpoint of any side of the hexagon.
    4. D. The point where the diagonals of the hexagon meet.
    [1]
11. 11.
    Construct a $\Delta PQR$, given $PQ = 5$ cm, $QR = 7$ cm and $\angle PQR = 60^\circ$. To inscribe a circle in the triangle, which step is required?

    1. A. Draw perpendicular bisectors of $PQ$ and $QR$ to find the centre.
    2. B. Draw angle bisectors of $\angle Q$ and $\angle R$ intersecting at $I$.
    3. C. Construct the circumcircle and use its centre for the incircle.
    4. D. Find the midpoint of $PR$ and use it as the centre of the incircle.
    [1]
12. 12.
    Using ruler and compasses only, draw an equilateral triangle of side 5 cm and draw its inscribed circle as shown in the diagram . Which labelled part in the diagram represents the radius of the inscribed circle?

    1. A. The perpendicular distance from the incentre to any side of the triangle
    2. B. The distance from the incentre to any vertex of the triangle
    3. C. The length of the angle bisector from a vertex to the opposite side
    4. D. The distance between two sides of the triangle along a median
    [1]
13. 13.
    Using ruler and compass only, draw a circle of radius 4 cm with centre O. Mark a point P outside the circle at a distance of 7 cm from O. Construct two tangents to the circle from P and measure the length of any one tangent. What is the length of one tangent?

    1. A. 4.9 cm
    2. B. 5.7 cm
    3. C. 6.3 cm
    4. D. 7.0 cm
    [1]
14. 14.
    Draw a circle of radius 3 cm. Construct tangents to the circle so that the angle between them is 45°. What is the measure of the angle subtended by the two points of tangency at the centre of the circle?

    1. A. 45°
    2. B. 90°
    3. C. 135°
    4. D. 180°
    [1]
15. 15.
    In the given diagram, , a circle of diameter 9 cm is drawn, and a point P is marked at a distance of 7.5 cm from the centre O of the circle. Tangents PA and PB are drawn from P to the circle. Which labelled segment correctly represents the length of each tangent?

    1. A. OA = 4.5 cm
    2. B. OP = 7.5 cm
    3. C. PA = 6 cm
    4. D. AB = 9 cm
    [1]
16. 16.
    (i) Using a ruler and compass, construct a ΔABC with AB = 6 cm, AC = 4.5 cm, and ∠BAC = 120°. After constructing the triangle, measure and determine the radius of the circumscribed circle around ΔABC.

    1. A. 3 cm
    2. B. 4.5 cm
    3. C. 6 cm
    4. D. 7.5 cm
    [1]
17. 17.
    Using a ruler, construct a ΔABC with BC = 6.4 cm, CA = 5.8 cm, and ∠ABC = 60°. Draw its incircle. What is the radius of this incircle?

    1. A. 1.0 cm
    2. B. 1.5 cm
    3. C. 2.0 cm
    4. D. 2.5 cm
    [1]
18. 18.
    In the given diagram , the point labelled H represents the orthocentre of triangle ABC. Which of the following correctly describes how this point is determined?

    1. A. It is the point where the perpendicular bisectors of the sides of the triangle meet.
    2. B. It is the point where the altitudes of the triangle, drawn from each vertex to the opposite side, intersect.
    3. C. It is the point where the angle bisectors of the triangle meet.
    4. D. It is the point where the medians of the triangle intersect.
    [1]
19. 19.
    Construct a triangle ABC, given that AB = 4 cm, BC = 6 cm, and median AM = 3 cm. After constructing the triangle, a circle is circumscribed about it. Where is the centre of this circumcircle located?

    1. A. At the intersection of the perpendicular bisectors of AB and BC
    2. B. At the intersection of the perpendicular bisectors of AB and AC
    3. C. At the intersection of the medians of the triangle
    4. D. At the intersection of the angle bisectors of the triangle
    [1]
20. 20.
    Draw a circle of radius 4 cm. To construct two tangents to the circle inclined at an angle of 60° to each other, what should be the angle between the two radii at the points of contact?

    1. A. 30°
    2. B. 60°
    3. C. 120°
    4. D. 180°
    [1]