Show that A(3, 2), B(0, 5), C(-3, 2) and D(0, -1) are the vertices of a square.
Given: The points are $A(3,2), B(0,5), C(-3,2)$ and $D(0,-1)$.
Note: For a quadrilateral to be a square, all the sides of the quadrilateral must be equal in length and the diagonals must be equal in length as well.
$A B=\sqrt{(0-3)^{2}+(5-2)^{2}}=\sqrt{9+9}$
$=3 \sqrt{2}$ units
$B C=\sqrt{(-3-0)^{2}+(2-5)^{2}}=\sqrt{9+9}$
$=3 \sqrt{2}$ units
$D A=\sqrt{(3-0)^{2}+(2+1)^{2}}=\sqrt{9+9}$
$=3 \sqrt{2}$ units
Therefore, AB = BC = CD = DA …..(1)
$A C=\sqrt{(-3-3)^{2}+(2-2)^{2}}$
$=6$ units
$B D=\sqrt{(0-0)^{2}+(-1-5)^{2}}$
$=6$ units
Therefore, $A C=B D \ldots .(2)$
From 1 and 2, we have all the sides of ABCD are equal and the diagonals are equal in length as well.
Therefore, $\mathrm{ABCD}$ is a square.
Hence, the points $A, B, C$ and $D$ are the vertices of a square.
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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,