1
A park is designed in the shape of a circle with a central fountain. The park's boundary is a path. A gardener observes that a straight path within the park, going from one point on the boundary to another, passes through the fountain.(a) What is the name of this straight path inside the park? [1 mark](b) If the straight path did not pass through the fountain but still connected two points on the boundary, what would it be called? [1 mark](c) If a straight path touches the boundary of the park at exactly one point, what is it called? 1mark
2
Case Study: A circular park has a pathway of uniform width around it. A security guard patrols the boundary of the park.(i) If the pathway is considered as the region between two concentric circles, and the guard walks along the outer edge of the pathway, what geometric shape does the guard's path represent? [1 mark](ii) If the guard walks along the inner edge of the pathway, which is tangent to the fountain at the center, what can be said about the radius to the point of contact? [1 mark](iii) Suppose the pathway is 2 meters wide, and the radius of the inner circle (fountain area) is 5 meters. A straight path connecting two points on the outer boundary passes through the center. What is the length of this path? 2marks
3
A student is drawing a circle and some lines. They draw a line that intersects the circle at exactly one point.(a) What is this line called? [1 mark](b) What is the relationship between the radius drawn to this point of intersection and the line? [1 mark](c) If the student draws another line parallel to the first one, but it intersects the circle at two points, what is this second line called? 1mark
4
Assertion (A): A tangent to a circle is a special case of a secant.Reason (R): A tangent intersects the circle at two distinct points, while a secant intersects at only one. 2marks
5
A cyclist is riding on a circular track. At any instant, the point where the tyre touches the ground is the point of contact.(a) What is the line representing the ground at the point of contact called with respect to the circular track? [1 mark](b) What is the relationship between the radius of the track passing through the point of contact and this line? [1 mark](c) If the cyclist falls, the bicycle slides in a straight line. What does this straight line represent in relation to the circular path the bicycle was on? 2marks
6
A wire is bent into a circular shape and then a straight segment is attached to it at a single point.(a) If the straight segment lies on the plane of the circle and touches the circle at only one point, it is called a _____. [1 mark](b) If the straight segment intersects the circle at two points, it is called a _____. [1 mark](c) If we consider the chord formed by the intersection of the secant with the circle, what happens to this chord as the secant moves towards becoming a tangent? 2marks
7
Case Study: Two tangents are drawn to a circle from an external point.Consider a point P outside a circle with center O. Let PA and PB be two tangents to the circle, where A and B are the points of contact.(i) What is the relationship between the lengths PA and PB? [1 mark](ii) What is the relationship between the angles ∠OPA and ∠OPB? [1 mark](iii) If ∠APB = 70°, find ∠OAP. 2marks
8
Assertion (A): The lengths of tangents drawn from an external point to a circle are equal.Reason (R): The two right-angled triangles formed by joining the external point to the center and the points of contact are congruent by RHS. 2marks
9
A circular pond has a bridge that crosses it. The bridge connects two points on the edge of the pond.(a) If the bridge is a straight structure and passes through the center of the pond, what is it called in relation to the circle? [1 mark](b) If the bridge is a straight structure but does not pass through the center, what is it called? [1 mark](c) If a straight edge of a floating platform touches the pond's edge at exactly one point, what is this edge called? 1mark
10
A craftsman is making a circular table. He needs to attach legs. The legs are placed at points on the circumference, and each leg is straight.(a) If a leg is placed such that it touches the circular edge at one point, what geometric term describes this leg's position relative to the table's edge at that point? [1 mark](b) If two legs are placed at the ends of a diameter, what can be said about the lines formed by these legs if they were extended infinitely? 2marks
11
Assertion (A): At any point on a circle, there can be only one tangent.Reason (R): A tangent is a line that intersects the circle at exactly one point, and the radius to this point is perpendicular to the tangent. 2marks
12
A farmer is fencing a circular field. He plans to use straight fences.(a) If a fence segment touches the circular boundary at only one point, what is it called? [1 mark](b) If a fence segment connects two points on the boundary and passes through the center, what is it called? [1 mark](c) If a fence segment connects two points on the boundary but does not pass through the center, what is it called? 1mark
13
Case Study: A circular garden has a central fountain. A straight pathway is built to connect two points on the outer edge of the garden.(i) If the pathway passes through the fountain, what is this pathway called with respect to the circular garden? [1 mark](ii) If the pathway touches the edge of the fountain (assumed to be a point at the center of the garden) at only one point, what is this pathway called with respect to the fountain? [1 mark](iii) If the pathway is a chord of the garden, and the radius to one endpoint of the pathway is drawn, what is the relationship between this radius and the tangent at that endpoint? 2marks
14
Assertion (A): In two concentric circles, the chord of the larger circle that touches the smaller circle is bisected at the point of contact.Reason (R): The radius of the smaller circle drawn to the point of contact is perpendicular to the chord. 2marks
15
A circular disc has a point marked on its edge. A straight ruler is placed against the disc.(a) If the ruler touches the disc at only the marked point, what is the ruler called with respect to the disc? [1 mark](b) If the ruler crosses the disc at two points, one of which is the marked point, what is the ruler called? [1 mark](c) If the ruler cuts the disc at two points, forming a line segment within the disc, what is this line segment called? 1mark
16
A building has a circular window. A straight rod is placed against the window.(a) If the rod touches the circular edge of the window at only one point, what is the rod called in relation to the window? [1 mark](b) If the rod passes through the center of the window and touches the edge at two points, what is the rod called? [1 mark](c) If the rod touches the window at one point, and a line segment is drawn from the center of the window to this point, what is the relationship between this line segment and the rod? 2marks
17
Case Study: A wheel of a vehicle moving on a road.(i) At any given moment, the point where the wheel touches the road is called the _____. [1 mark](ii) The road at this point is ____ to the circular wheel. [1 mark](iii) The radius drawn from the center of the wheel to this point of contact is ____ to the road. [1 mark](iv) If the vehicle suddenly skids and moves in a straight line, this line represents a _____ to the wheel. 2marks
18
Assertion (A): The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.Reason (R): A quadrilateral formed by the external point, the centre, and the two points of contact is a cyclic quadrilateral. 2marks
19
A craftsman is engraving a circular design. He makes a straight cut that passes through the design.(a) If the cut touches the boundary of the design at only one point, what is this cut called? [1 mark](b) If the cut passes through the center and touches the boundary at two points, what is this cut called? [1 mark](c) If the cut passes through the interior of the design and connects two points on the boundary, what is this cut called? 1mark
20
Case Study: Two tangents TP and TQ are drawn to a circle with centre O from an external point T.(i) Prove that TP = TQ. [3 marks](ii) If ∠PTQ = 60°, what type of triangle is ΔTPQ? Justify your answer. 2marks
Answers
1.(a) Diameter [1 mark](b) Chord [1 mark](c) Tangent 1mark
2.
(i) Circle [1 mark]
(ii) The radius is perpendicular to the tangent (pathway edge). [1 mark]
(iii) The pathway passes through the center, so it's a diameter. Radius of inner circle = 5m. Width of pathway = 2m. Radius of outer circle = 5 + 2 = 7m. Diameter of outer circle = 2 * 7 = 14m. [2 marks]
3.
(a) Tangent [1 mark]
(b) The radius to the point of intersection is perpendicular to the tangent. [1 mark]
(c) Secant [1 mark]
4
Assertion is True, Reason is False. A tangent intersects at only one point. 2marks
5.
(a) Tangent [1 mark]
(b) The radius to the point of contact is perpendicular to the tangent (ground). [1 mark]
(c) Tangent [2 marks]
6.
(a) Tangent [1 mark]
(b) Secant [1 mark]
(c) The chord gradually decreases in length and becomes a point when the secant becomes a tangent. [2 marks]
7.
(i) PA = PB (The lengths of tangents drawn from an external point to a circle are equal). [1 mark]
(ii) ∠OPA = ∠OPB (OP bisects ∠APB and also is the line connecting the center to the external point, forming congruent triangles). [1 mark]
(iii) In right-angled triangle OAP, ∠OAP + ∠OPA = 90°. Since ∠APB = 70°, then ∠OPB = ∠OPA = 70°/2 = 35°. So, ∠OAP = 90° - 35° = 55°. [2 marks]
8
Assertion is True, Reason is True. 2marks
9.
(a) Diameter [1 mark]
(b) Chord [1 mark]
(c) Tangent [1 mark]
10.
(a) Tangent [1 mark]
(b) The lines formed by extending the legs would be parallel tangents. [2 marks]
11
Assertion is True, Reason is True. 2marks
12.
(a) Tangent [1 mark]
(b) Diameter [1 mark]
(c) Chord [1 mark]
13.
(i) Diameter [1 mark]
(ii) Tangent [1 mark]
(iii) The radius is perpendicular to the tangent. [2 marks]
14
Assertion is True, Reason is True. 2marks
15.
(a) Tangent [1 mark]
(b) Secant [1 mark]
(c) Chord [1 mark]
16.
(a) Tangent [1 mark]
(b) Diameter [1 mark]
(c) The line segment (radius) is perpendicular to the rod (tangent) at the point of contact. [2 marks]
17.
(i) Point of contact [1 mark]
(ii) Tangent [1 mark]
(iii) Perpendicular [1 mark]
(iv) Tangent [2 marks]
18
Assertion is True, Reason is True. 2marks
19.
(a) Tangent [1 mark]
(b) Diameter [1 mark]
(c) Chord [1 mark]
20.(i) Proof: Consider ΔTPO and ΔTQO.
1
TP and TQ are tangents from external point T.
2
OP = OQ (radii of the same circle).
3
OT is common.
4
∠TPO = ∠TQO = 90° (radius is perpendicular to the tangent at the point of contact).Therefore, by RHS congruence rule, ΔTPO ≅ ΔTQO.Hence, TP = TQ (CPCT). [3 marks](ii) If ∠PTQ = 60°, and we know TP = TQ, then ΔTPQ is an isosceles triangle with the angle between the equal sides being 60°.In an isosceles triangle, the angles opposite the equal sides are equal: ∠TPQ = ∠TQP.The sum of angles in a triangle is 180°.So, ∠TPQ + ∠TQP + ∠PTQ = 180°2 * ∠TPQ + 60° = 180°2 * ∠TPQ = 120°∠TPQ = 60°Since all three angles are 60°, ΔTPQ is an equilateral triangle. 2marks
