SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]
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4 Marks
Q1. Case Study: Shadow of a Lamp-Post & Building Height
A street light bulb is fixed on a pole $6\text{ m}$ above the level ground. A girl of height $1.5\text{ m}$ is walking away from the base of the lamp-post at a uniform speed of $1.2\text{ m/s}$.
a) State the similarity criterion used to establish similarity between the triangle formed by the pole and its shadow and the triangle formed by the girl and her shadow (1 Marks)
b) If after $4\text{ seconds}$, the girl is $4.8\text{ m}$ away from the base of the pole, find the length of her shadow. (2 Marks)
c) Find the total distance of the tip of her shadow from the base of the lamp-post at that instant (1 Marks)
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4 Marks
Q2. Case Study: Bridge Truss Design & Triangular Elements
An engineer is designing a steel bridge truss structure. One of the main supporting segments consists of $\triangle PQR$ where a line segment $ST$ is drawn parallel to $QR$ intersecting $PQ$ at $S$ and $PR$ at $T$.
a) State the theorem used to relate the proportional segments created by line $ST$ parallel to $QR$ (1 Marks)
b) If $\frac{PS}{SQ} = \frac{3}{4}$ and $PR = 28\text{ m}$, find the length of $PT$ (2 Marks)
c) If $ST = 6\text{ m}$, find the length of $QR$ given that $\triangle PST \sim \triangle PQR$. (1 Marks)
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4 Marks
Q3. Case Study: A Park Layout & Similar Triangular Lawns
A city municipal corporation plans to design a triangular public park shaped like $\triangle ABC$ such that another smaller triangular lawn $\triangle PQR$ is laid inside it with corresponding sides proportional. It is given that $\triangle ABC \sim \triangle PQR$.
a) If the ratio of the corresponding sides $\frac{AB}{PQ} = \frac{2}{3}$, write the ratio of their corresponding perimeters (1 Marks)
b) If the area of $\triangle PQR$ is $135\text{ cm}^2$, find the area of $\triangle ABC$. (2 Marks)
c) If altitude $AD$ of $\triangle ABC$ is $4\text{ cm}$, find the corresponding altitude $PS$ of $\triangle PQR$ (1 Marks)
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4 Marks
Q4. Case Study: Shadow of a Vertical Tower vs. Pole
During a school mathematics project, students measure shadows in the playground. A vertical pole of length $6\text{ m}$ casts a shadow $4\text{ m}$ long on the ground. At the same time, a nearby cellular tower casts a shadow $28\text{ m}$ long.
a) Name the similarity criterion that proves the triangles formed by the sun's rays, objects, and their shadows are similar. (1 Marks)
b) Formulate a proportional equation relating the heights and shadows of the pole and the tower (1 Marks)
c) Calculate the height of the tower (2 Marks)
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4 Marks
Q5. Case Study: Trapezium-shaped Land Parcel
A real estate surveyor measures a plot of land shaped like a trapezium $ABCD$ where $AB \parallel DC$. The diagonals $AC$ and $BD$ intersect at point $O$
a) State the similarity criterion applicable to $\triangle AOB$ and $\triangle COD$ (1 Marks)
b) Prove the ratio relation $\frac{AO}{OC} = \frac{BO}{OD}$ (2 Marks)
c) If $AB = 2CD$ and the area of $\triangle COD = 20\text{ m}^2$, find the area of $\triangle AOB$. (1 Marks)
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4 Marks
Q6. Case Study: Ladder Against a Building Wall
A painter places a $10\text{ m}$ long ladder against a vertical wall such that its foot is $6\text{ m}$ away from the base of the wall on level ground. Later, the top slips down a bit.
a) Name the theorem used to find the height reached by the ladder on the wall. (1 Marks)
b) Find the initial height of the window where the top of the ladder reaches. (2 Marks)
c) If the foot of the ladder is pulled $2\text{ m}$ further away from the wall, find the new height reached by the top of the ladder. (1 Marks)
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4 Marks
Q7. Case Study:Triangular Roof Truss and Altitudes
An architect designs a roof truss in the shape of a right-angled triangle $\triangle ABC$ where $\angle B = 90^\circ$. A perpendicular $BD$ is drawn from the right-angled vertex $B$ to the hypotenuse $AC$.
a) State the similarity relation between $\triangle ADB$ and $\triangle BDC$. (1 Marks)
b) Prove that $BD^2 = AD \times CD$. (2 Marks)
c) If $AD = 4\text{ cm}$ and $CD = 9\text{ cm}$, find the length of $BD$ (1 Marks)
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4 Marks
Q8. Case Study: Field Surveying & Mid-Point Division
A surveyor marks three points $A, B$, and $C$ on a map representing three water reservoirs. Points $D$ and $E$ are marked as the mid-points of sides $AB$ and $AC$ respectively, and line segment $DE$ is joined.
a) State the Basic Proportionality Theorem (Thales' Theorem) (1 Marks)
b) Prove that $DE \parallel BC$ (2 Marks)
c) If $BC = 12\text{ cm}$, find the length of $DE$ (1 Marks)
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4 Marks
Q9. Case Study: Triangular Garden Paths
A triangular garden $\triangle ABC$ has a walking track constructed parallel to side $BC$ intersecting side $AB$ at $D$ and side $AC$ at $E$
a) If $\frac{AD}{DB} = \frac{2}{3}$ and $AC = 15\text{ m}$, find the length of $AE$ (2 Marks)
b) Name the theorem applied to arrive at this result. (1 Marks)
c) If $DE = 4\text{ m}$, find the length of $BC$ given $\triangle ADE \sim \triangle ABC$ (1 Marks)
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4 Marks
Q10. Case Study: Street Light Illumination Geometry
Two street light poles of heights $a$ and $b$ stand $d$ meters apart on opposite sides of a straight road. The lines joining the top of each pole to the foot of the opposite pole intersect at a point $P$ above the ground.
a) Draw or visualize the intersecting triangles and identify the pair of similar triangles sharing the height of intersection $h$. (1 Marks)
b) Formulate the reciprocal height relation $\frac{1}{h} = \frac{1}{a} + \frac{1}{b}$. (2 Marks)
c) If $a = 6\text{ m}$ and $b = 12\text{ m}$, find the height $h$ above the ground where the light beams cross (1 Marks)
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4 Marks
Q11. Case Study: Triangular Land Division
A farmer divides a triangular plot $\triangle PQR$ by drawing a line segment $LM$ parallel to $QR$, where $L$ lies on $PQ$ and $M$ lies on $PR$.
a) If $PM = 2\text{ cm}$, $MQ = 3\text{ cm}$, and $PL = 4\text{ cm}$, find the length of $LR$ (2 Marks)
b) State the geometric theorem supporting your calculation. (1 Marks)
c) If the ratio of the area of $\triangle PLM$ to $\triangle PQR$ is known, what is the ratio of their corresponding altitudes? (1 Marks)
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4 Marks
Q12. Case Study: Architectural Model Scale Comparison
An architectural firm builds two similar scale models of a triangular glass pavilion, Model-1 and Model-2. The ratio of their corresponding side lengths is $3:5$.
a) State the theorem governing the ratio of areas of two similar triangles (1 Marks)
b) Find the ratio of the surface areas of Model-1 to Model-2. (2 Marks)
c) If the volume scaling follows length cubes, and Model-1 has a base area of $45\text{ cm}^2$, find the base area of Model-2. (1 Marks)
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4 Marks
Q13. Case Study: Surveying a River Obstacle
To find the width of a river, a surveyor sets up triangulation markers. Two similar triangles $\triangle ABC$ and $\triangle PQR$ are formed across the banks, with scale factor $\frac{AB}{PQ} = \frac{2}{5}$.
a) If the perimeter of $\triangle ABC$ is $24\text{ m}$, find the perimeter of $\triangle PQR$. (2 Marks)
b) State the similarity criterion used when all three corresponding side ratios are equal (1 Marks)
c) If side $BC = 10\text{ m}$, find the corresponding side $QR$. (1 Marks)
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4 Marks
Q14. Case Study: Rhombus Diagonal Properties
A decorative metal frame is manufactured in the shape of a rhombus $ABCD$ with side length $10\text{ cm}$. One of its diagonals is $16\text{ cm}$ long.
a) State the angle property at which the diagonals of a rhombus intersect each other. (1 Marks)
b) Use Pythagoras theorem in the right-angled triangle formed at the intersection point to find half of the second diagonal (2 Marks)
c) Find the total length of the second diagonal. (1 Marks)
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4 Marks
Q15. Case Study: Shadow of an Electric Pole
An electric pole casts a shadow of length $20\text{ m}$ when the angle of elevation of the sun is such that a nearby $2\text{ m}$ vertical stick casts a shadow of $1.5\text{ m}$
a) Name the similarity criterion for the triangles formed by the sun rays, poles, and shadows. (1 Marks)
b) Formulate the proportion between the pole height and stick height (2 Marks)
c) Calculate the height of the electric pole (1 Marks)
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4 Marks
Q16. Case Study: Right Triangle Altitude Proportions
A triangular metallic plate $\triangle ABC$ is right-angled at $A$, and an altitude $AD$ is drawn to the hypotenuse $BC$.
a) Prove that $\triangle DBA \sim \triangle DAC$. (2 Marks)
b) Write the ratio of sides equating $AD$ segments. (1 Marks)
c) If $BD = 2\text{ cm}$ and $DC = 8\text{ cm}$, find the length of $AD$. (1 Marks)
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4 Marks
Q17. Case Study: Equilateral Triangle Internal Point
In an equilateral triangle $ABC$, $D$ is a point on side $BC$ such that $BD = \frac{1}{3}BC$.
a) Draw an altitude from $A$ to $BC$ meeting at $E$, and express segments $BE$ and $DE$ in terms of side $a$ (1 Marks)
b) Apply Pythagoras theorem to $\triangle ADE$ and $\triangle ABD$ (2 Marks)
c) Prove that $9AD^2 = 7AB^2$. (1 Marks)
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4 Marks
Q18. Case Study: Flagstaff on a Building Roof
A flagstaff is mounted on top of a building. From a point on the ground $50\text{ m}$ away from the building base, the angles of elevation of the top and bottom of the flagstaff are observed, forming similar right-angled triangle scaling heights.
a) If the building height is $h_1$ and flagstaff height is $h_2$, state the trigonometric/geometric similarity principle involved. (1 Marks)
b) If the ratio of two right triangles' bases and heights are proportional, write the proportional relation (2 Marks)
c) Calculate the height scaling if base distance scales by a factor of $1.5$. (1 Marks)
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4 Marks
Q19. Case Study: Triangular Park Fencing Layout
A triangular park has sides measuring $6\text{ m}, 8\text{ m}$, and $10\text{ m}$. A smaller similar triangular flower bed is created inside with perimeter equal to half of the outer park.
a) State the ratio of the perimeters of two similar triangles. (1 Marks)
b) Find the lengths of the sides of the inner flower bed (2 Marks)
c) Find the ratio of the area of the inner flower bed to the outer park (1 Marks)
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4 Marks
Q20. Case Study: Intersection of Diagonals in a Trapezium
In a trapezium $ABCD$ with $AB \parallel DC$, diagonals $AC$ and $BD$ intersect at point $O$. Through $O$, a line parallel to $AB$ is drawn meeting $AD$ at $E$ and $BC$ at $F$.
a) Using BPT in $\triangle DAB$, express $\frac{DE}{EA}$ in terms of segment ratios. (1 Marks)
b) Using BPT in $\triangle CAB$, relate segment $OF$ (2 Marks)
c) Prove that $EO = OF$ (1 Marks)
Academic Repository Disclaimer & எச்சரிக்கை அறிவிப்பு :
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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