In fig. (10).23, PQRS is a square and SRT is an equilateral triangle. Prove that (i) PT = QT (ii) ∠TQR = 15°
In fig. (10).23, PQRS is a square and SRT is an equilateral triangle. Prove that (i) PT = QT (ii) ∠TQR = 15°
Given that PQRS is a square and SRT is an equilateral triangle. And given to prove that
(i) PT = QT and (ii) ∠TQR = 15°
Now, PQRS is a square
PQ = QR = RS = SP ... (i)
And ∠SPQ = ∠PQR = ∠QRS = ∠RSP = 90° = right angle
And also, SRT is an equilateral triangle.
SR = RT = TS ... (ii)
And ∠TSR = ∠SRT = ∠RTS = 60°
From (i) and (ii)
PQ = QR = SP = SR = RT = TS .... (iii)
And also,
∠TSP = ∠TSR + ∠RSP = 60° + 90° + 150°
∠TRQ = ∠TRS + ∠SRQ = 60° + 90° + 150°
⟹ ∠TSR = ∠TRQ = 150° ... (iv)
SP = RQ [From (iii)]
So, by SAS congruence criterion we have
ΔTSP = ΔTRQ
PT = QT [Corresponding parts of congruent triangles are equal] Consider ΔTQR.
QR = TR [From (iii)]
ΔTQR is a isosceles triangle.
∠QTR = ∠TQR [angles opposite to equal sides]
Now,
Sum of angles in a triangle is equal to 180∘
⟹ ∠QTR + ∠TQR + ∠TRQ = 180°
⟹ 2∠ TQR + 150° = 180° [From (iv)]
⟹ 2∠TQR = 180° - 150°
⟹ 2∠ TQR = 30° ∠TQR = 15°] ...
Hence proved
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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,