In a ∆ABC, point D is on side AB and point E is on side AC, such that BCED is a trapezium. If DE : BC = 3 : 5, then Area (∆ ADE) : Area (◻BCED) =
In a ∆ABC, point D is on side AB and point E is on side AC, such that BCED is a trapezium. If DE : BC = 3 : 5, then Area (∆ ADE) : Area (◻BCED) =
(a) 3 : 4
(b) 9 : 16
(c) 3 : 5
(d) 9 : 25
Given: In ΔABC, D is on side AB and point E is on side AC, such that BCED is a trapezium. DE: BC = 3:5.
To find: Calculate the ratio of the areas of ΔADE and the trapezium BCED.
In ΔADE and ΔABC,
∠ADE=∠B Corresponding angles∠A=∠A Common∴∆ADE~∆ABC AA Similarity
$\frac{\operatorname{Ar}(\Delta \mathrm{ADE})}{\operatorname{Ar}(\Delta \mathrm{ABC})}=\frac{\mathrm{DE}^{2}}{\mathrm{BC}^{2}}$
$\frac{\operatorname{Ar}(\triangle \mathrm{ADE})}{\operatorname{Ar}(\triangle \mathrm{ABC})}=\frac{3^{2}}{5^{2}}$
$\frac{\operatorname{Ar}(\triangle \mathrm{ADE})}{\operatorname{Ar}(\triangle \mathrm{ABC})}=\frac{9}{25}$
Let Area of ΔADE = 9x sq. units and Area of ΔABC = 25x sq. units
$\operatorname{Ar}[\operatorname{trapBCED}]=\operatorname{Ar}(\triangle \mathrm{ABC})-\operatorname{Ar}(\triangle \mathrm{ADE})$
$=25 x-9 x$
$=16 x \mathrm{sq}$ units
Now ,
$\frac{\operatorname{Ar}(\triangle \mathrm{ADE})}{\operatorname{Ar}(\operatorname{trapBCED})}=\frac{9 x}{16 x}$
$\frac{\operatorname{Ar}(\triangle \mathrm{ADE})}{\operatorname{Ar}(\operatorname{trapBCED})}=\frac{9}{16}$
Hence the correct answer is $(b)$.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,