If the equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, – 1), then find the length of the side of the triangle.
Let $\triangle \mathrm{ABC}$ be an equilateral triangle.
Given equation of the base $B C$ is $x+y=2$
We know that, in an equilateral triangle all angles are of $60^{\circ}$
So, in $\triangle \mathrm{ABD}$
$\sin 60^{\circ}=\frac{\mathrm{AD}}{\mathrm{AB}}$\
$\Rightarrow \frac{\sqrt{3}}{2}=\frac{\mathrm{AD}}{\mathrm{AB}}\left[\because \sin 60^{\circ}=\frac{\sqrt{3}}{2}\right]$
$\Rightarrow \mathrm{AD}=\frac{\sqrt{3}}{2} \mathrm{AB}$
We know that, the distance $d$ of a point $P\left(x_{0}, y_{0}\right)$ from the line $A x+B y+C=0$ is given by
$d=\left|\frac{A x_{0}+B y_{0}+C}{\sqrt{A^{2}+B^{2}}}\right|$
Now, length of perpendicular from vertex $A(2,-1)$ to the line $x+y=2$ is
$\mathrm{AD}=\left|\frac{1 \times 2+1 \times(-1)-2}{\sqrt{(1)^{2}+(1)^{2}}}\right|$
$\Rightarrow \frac{\sqrt{3}}{2} \mathrm{AB}=\left|\frac{2-1-2}{\sqrt{2}}\right|$
On simplification we get
$\Rightarrow \frac{\sqrt{3}}{2} \mathrm{AB}=\frac{1}{\sqrt{2}}$
Squaring both the sides, we get
$\Rightarrow \frac{3}{4} \mathrm{AB}^{2}=\frac{1}{2}$
On cross multiplication we get
$\Rightarrow \mathrm{AB}^{2}=\frac{4}{3} \times \frac{1}{2}$
$\Rightarrow \mathrm{AB}^{2}=\frac{2}{3}$
$\Rightarrow A B=\sqrt{\frac{2}{3}}$
Hence, the required length of side is $\sqrt{\frac{2}{3}}$
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- 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,