SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]
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4 Marks
Q1. Case Study: The Circular Park and Fountain
A municipal corporation designs a circular park with a central fountain. The park has a radius of $15\text{ metres}$. Two straight walking paths $PA$ and $PB$ are constructed from an entry gate $P$ outside the park such that they act as tangents to the circular boundary at points $A$ and $B$ respectively. The distance of the entry gate $P$ from the center $O$ of the park is $25\text{ metres}$.
a) Find the length of the tangent path $PA$ from the entry gate to the point of contact on the park boundary. (1 Marks)
b) State the geometrical relation between the radius $OA$ and the tangent path $PA$ at the point of contact $A$. (1 Marks)
c) If another path connects the points of contact $A$ and $B$, find the length of chord $AB$ in terms of the radius and distance $OP$. (2 Marks)
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4 Marks
Q2. Case Study: The Ferris Wheel Illumination
During a local festival, a giant circular Ferris wheel of radius $20\text{ metres}$ is set up. The ride rotates slowly around its center $O$. A spectator standing at a viewing gallery on the circumference at point $C$ watches two illuminated passenger capsules positioned at points $A$ and $B$ on the wheel's rim. The arc $AB$ subtends an angle of $70^\circ$ at the center $O$.
a) What will be the angle subtended by the arc $AB$ at the spectator's viewing position $C$ on the remaining part of the circle? (1 Marks)
b) If capsule $A$ moves along the rim such that arc $AB$ doubles its length at the center, what is the new angle at the center? (1 Marks)
c) Prove that if chord $AB$ equals the radius of the wheel, the angle subtended by chord $AB$ at any point in the major segment is $30^\circ$. (2 Marks)
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4 Marks
Q3. Case Study: The Arch Bridge Design
An engineer designs an arch bridge supported by a large steel circular arc. The span of the bridge chord $AB$ is $24\text{ metres}$, and the radius of the circular arch is $13\text{ metres}$. A vertical support pillar $OM$ is dropped from the center $O$ of the circle to the chord $AB$.
a) State the property related to the perpendicular dropped from the center of a circle to a chord (1 Marks)
b) Calculate the height of the support pillar $OM$ from the center to the bridge chord. (1 Marks)
c) If another parallel bridge chord of length $10\text{ metres}$ is constructed on the same circular arch, find the distance of this new chord from the center. (2 Marks)
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4 Marks
Q4. Case Study: The Circular Traffic Roundabout
A city traffic planner designs a circular roundabout of radius $7\text{ metres}$. Four approach roads meet the roundabout perimeter, forming tangents at points $P, Q, R,$ and $S$. Two primary intersecting tangents from an external control tower $T$ touch the roundabout at points $A$ and $B$ such that $\angle ATB = 60^\circ$.
a) Find the length of the line segment joining the control tower $T$ to the center $O$ of the roundabout. (1 Marks)
b) What is the sum of angles $\angle TAO$ and $\angle TBO$? (1 Marks)
c) Find the area of the quadrilateral $TAOB$ formed by the tangents and radii meeting at the points of contact. (2 Marks)
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4 Marks
Q5. Case Study: The Stained Glass Window
An artist creates a decorative circular stained-glass window containing a cyclic quadrilateral $ABCD$ inscribed within the glass frame. The opposite angles of the quadrilateral are designed such that $\angle A = (3x - 10)^\circ$ and $\angle C = (x + 30)^\circ$.
a) State the fundamental theorem applicable to the opposite angles of a cyclic quadrilateral. (1 Marks)
b) Solve for the value of $x$ based on the given angle expressions (1 Marks)
c) Calculate the exact numerical measures of angles $\angle A$ and $\angle C$, and determine if the quadrilateral is a rectangle. (2 Marks)
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4 Marks
Q6. Case Study: The Circular Running Track and Coach
A sports academy features two concentric circular running tracks with radii $8\text{ metres}$ and $5\text{ metres}$ respectively. A coach standing on the outer track watches a runner sprint along a straight track segment $AB$ that acts as a chord of the outer track and touches the inner track tangentially at point $P$.
a) What is the perpendicular distance from the center to the running path $AB$ touching the inner track? (1 Marks)
b) State the geometric property used to relate the radius of the inner circle to the outer chord. (1 Marks)
c) Calculate the total length of the running chord $AB$ on the outer track (2 Marks)
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4 Marks
Q7. Case Study:The Billboard Structure
An outdoor advertising agency erects a triangular billboard mounting platform supported by two steel cables $PA$ and $PB$ anchored from an observation deck $P$ to a cylindrical pillar of radius $6\text{ metres}$ at points $A$ and $B$. The length of cable $PA$ is $8\text{ metres}$.
a) State the theorem concerning the lengths of tangents drawn from an external point to a circle. (1 Marks)
b) Determine the length of the second cable $PB$ (1 Marks)
c) If the distance from the observation deck $P$ to the center $O$ of the cylindrical pillar is $10\text{ metres}$, verify the right-angled triangle relation for $\triangle OAP$. (2 Marks)
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4 Marks
Q8. Case Study: The Circular Logo Design
A graphic designer is creating a brand logo featuring a circle with a tangent line $XY$ touching at point $P$, and a chord $PQ$ drawn from the point of contact. An angle $\theta$ is formed between the tangent $XY$ and the chord $PQ$.
a) Name the theorem that relates the angle between a tangent and a chord to the angles in the circle. (1 Marks)
b) If the angle between the tangent and chord is $55^\circ$, what is the measure of the angle subtended by chord $PQ$ in the alternate segment? (1 Marks)
c) Explain how this property is used to prove concyclicity when constructing inscribed triangles in the logo. (2 Marks)
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4 Marks
Q9. Case Study: The Triangular Land Plot
A real estate developer plans a triangular park $\triangle ABC$ with side lengths $AB = 9\text{ cm}$, $BC = 12\text{ cm}$, and $AC = 15\text{ cm}$ (scaled in meters). A circular water fountain is inscribed inside the park, touching the sides $AB, BC,$ and $AC$ at points $D, E,$ and $F$ respectively.
a) State the relation between the tangent segments drawn from any vertex of the triangle to the inscribed circle. (1 Marks)
b) If the length of tangent segment from vertex $B$ to the contact point $D$ is $x$, express the lengths of the remaining segments. (1 Marks)
c) Calculate the radius of the inscribed circular water fountain. (2 Marks)
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4 Marks
Q10. Case Study: The Satellite Dish
A telecommunications engineer models a satellite signal path as a secant line passing through a circular receiver dish. A tangent segment $PT$ of length $12\text{ cm}$ is drawn from an external relay point $P$ to the rim of the circular dish, and a secant line $PAB$ passes through the center.
a) State the secant-tangent theorem product relation for point $P$. (2 Marks)
b) If the external segment $PA$ of the secant is $8\text{ cm}$, find the length of the entire secant segment $PB$. (2 Marks)
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4 Marks
Q11. Case Study: The Planetary Orbit Model
In an astronomy lab simulation, two planetary motion tracking chords $AB$ and $CD$ of a circular orbit intersect each other at right angles inside the orbital path at point $M$.
a) Define what a chord of a circle represents in terms of interior distance (1 Marks)
b) If the perpendicular distance from the center to chord $AB$ is $4\text{ cm}$ and the radius is $5\text{ cm}$, find the length of chord $AB$. (1 Marks)
c) Discuss how intersecting chords maintain proportional segment products ($AM \cdot MB = CM \cdot MD$) (2 Marks)
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4 Marks
Q12. Case Study: The Merry-Go-Round Safety Barrier
A children's playground features a rhombic safety barrier $ABCD$ built around a circular merry-go-round base.
a) Prove or state whether a parallelogram circumscribing a circle is always a rhombus. (1 Marks)
b) If side $AB$ of the barrier is $7\text{ cm}$, what is the length of side $BC$? (1 Marks)
c) If the perimeter of the barrier is $40\text{ cm}$, find the exact length of each individual side of the safety barrier. (2 Marks)
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4 Marks
Q13. Case Study: The Clock Face Geometry
An antique clock face is circular. The minute hand moves along the circumference. At 4:00 o'clock, the angle formed at the center by the hour and minute hands is $120^\circ$.
a) Find the angle subtended by this arc at any point on the circumference of the clock face. (1 Marks)
b) State the theorem linking the center angle to the circumference angle for the same arc. (1 Marks)
c) If the tip of the minute hand traces an arc that subtends $60^\circ$ at the center, calculate the ratio of the arc length to the circumference of the clock face. (2 Marks)
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4 Marks
Q14. Case Study: The Radar Tracking Station
A coastal radar station $P$ tracks a ship. Two radar tangent beams $PA$ and $PB$ are sent to a circular island of radius $r$. The angle between the two radar tangents is $\angle APB = 70^\circ$.
a) State the relationship between the angle between tangents ($\angle APB$) and the angle subtended at the center ($\angle AOB$). (1 Marks)
b) Calculate the measure of angle $\angle AOB$ (1 Marks)
c) If $\angle OPA$ is calculated, show that $\angle OPA = 55^\circ$ using triangle congruence properties. (2 Marks)
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4 Marks
Q15. Case Study: The Suspension Bridge Tower
A suspension bridge has two vertical towers with parallel vertical tangents $XY$ and $X'Y'$ to a circular cable curvature of radius $r = 10\text{ metres}$. A third inclined cable tangent intersects these parallel tangents at points $A$ and $B$.
a) State the property of the angle subtended by the tangent intercept $AB$ at the center of the circle (1 Marks)
b) What is the measure of angle $\angle AOB$ formed at the center by the point of contact $C$ of the inclined cable? (1 Marks)
c) Explain why the tangent segment between two parallel tangents always subtends a right angle at the center. (2 Marks)
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4 Marks
Q16. Case Study: The Circular Fountain Basin
A landscape architect designs a stone pathway around a circular fountain basin. The stone borders form a quadrilateral $ABCD$ that completely circumscribes the circular fountain.
a) Write the mathematical equation relating the opposite sides of quadrilateral $ABCD$ circumscribing the circle. (1 Marks)
b) If side $AB = 8\text{ m}$, $BC = 7\text{ m}$, and $CD = 6\text{ m}$, find the length of the fourth side $AD$. (1 Marks)
c) Verify the perimeter property if the sides are extended into tangential segments from the four outer corners. (2 Marks)
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4 Marks
Q17. Case Study: The Circular Stage Design
A theatre stage is built in the shape of a cyclic trapezium inscribed within a circular boundary. The non-parallel sides of the trapezium are designed to be equal in length to maintain symmetry.
a) State the condition under which a trapezium is guaranteed to be cyclic. (1 Marks)
b) If one interior angle of the cyclic trapezium is $70^\circ$, find the measure of its adjacent angle on the same side. (1 Marks)
c) Calculate the opposite angle corresponding to the $70^\circ$ angle using cyclic quadrilateral properties. (2 Marks)
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4 Marks
Q18. Case Study: The Archery Target
An archery target consists of concentric circular rings. The bullseye has a radius of $3\text{ cm}$ and the outer scoring ring has a radius of $5\text{ cm}$. A stray arrow hits the outer ring, acting as a chord of the outer circle that touches the inner bullseye circle tangentially.
a) State the relationship between the radius of the inner circle and the perpendicular distance from the center to the chord. (1 Marks)
b) Calculate the half-length of the chord using the radii of the two concentric circles. (1 Marks)
c) Find the total length of the chord (arrow path) across the outer scoring ring. (2 Marks)
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4 Marks
Q19. Case Study: The Circular Tunnel Cross-Section
An engineering team inspects a circular drainage tunnel of radius $r$. A metal safety rod is placed along the diameter $AB$. A monitoring sensor is placed at any point $C$ on the upper wall of the tunnel perimeter.
a) State the geometric theorem concerning the angle subtended by a diameter at any point on the circle. (1 Marks)
b) What is the exact measure of $\angle ACB$ formed by the sensor at point $C$? (1 Marks)
c) If $AC = 6\text{ cm}$ and $BC = 8\text{ cm}$, calculate the diameter of the circular tunnel using the Pythagorean theorem. (2 Marks)
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4 Marks
Q20. Case Study: The Circular Roundabout Landscaping
A city planner places a decorative statue at the center $O$ of a circular roundabout. A straight inspection path is laid from a lighting pole $P$ outside the roundabout such that the length of the tangent path is $24\text{ metres}$, and the distance from pole $P$ to the roundabout center $O$ is $26\text{ metres}$.
a) Identify the right-angled triangle formed by the radius, tangent, and center distance. (1 Marks)
b) Apply the Pythagorean theorem to set up the equation for the radius of the roundabout. (1 Marks)
c) Calculate the exact radius of the circular roundabout. (2 Marks)
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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