3. If (2,4),(2,6) are two vertices of an equilateral triangle then the third vertex is
Answers
Answer:
The coordinates of the third vertex of the triangle are ( 2 ± √3, 5 ).
Step-by-step-explanation:
We have given the coordinates of an equilateral triangle.
Let the triangle be ABC.
- A ≡ ( 2, 4 ) ≡ ( x₁, y₁ )
- B ≡ ( 2, 6 ) ≡ ( x₂, y₂ )
- C ≡ ( x, y )
Now, we know that,
All three sides of an equilateral triangle are congruent.
∴ AB = BC = AC
Now, by distance formula,
d ( A, C ) = d ( B, C )
⇒ √[ ( x₁ - x )² + ( y₁ - y )² ] = √[ ( x₂ - x )² + ( y₂ - y )² ]
By squaring both sides, we get,
⇒ ( x₁ - x )² + ( y₁ - y )² = ( x₂ - x )² + ( y₂ - y )²
⇒ ( 2 - x )² + ( 4 - y )² = ( 2 - x )² + ( 6 - y )²
⇒ ( 4 - y )² = ( 6 - y )²
⇒ ( 4 )² - 2 * 4 * y + y² = ( 6 )² - 2 * 6 * y + y²
⇒ 16 - 8y + y² = 36 - 12y + y²
⇒ 16 - 8y = 36 - 12y
⇒ - 8y + 12y = 36 - 16
⇒ 4y = 20
⇒ y = 20 ÷ 4
⇒ y = 5
Now, by distance formula,
d ( A, B ) = d ( A, C )
⇒ √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ] = √[ ( x₁ - x )² + ( y₁ - y )² ]
By squaring both sides, we get,
⇒ ( x₁ - x₂ )² + ( y₁ - y₂ )² = ( x₁ - x )² + ( y₁ - y )²
⇒ ( 2 - 2 )² + ( 4 - 6 )² = ( 2 - x )² + ( 4 - 5 )²
⇒ ( 0 )² + ( - 2 )² = ( 2 )² - 2 * 2 * x + x² + ( - 1 )²
⇒ 0 + 4 = 4 - 4x + x² + 1
⇒ 0 = x² - 4x + 1
⇒ x² - 4x + 1 = 0
⇒ x² - 4x + 1 + 3 - 3 = 0
⇒ x² - 4x + 4 - 3 = 0
⇒ ( x - 2 )² - 3 = 0
⇒ ( x - 2 )² = 3
⇒ ( x - 2 ) = ± √3
⇒ x = ± √3 + 2
⇒ x = 2 ± √3
∴ The coordinates of the third vertex of the triangle are ( 2 ± √3, 5 ).
Step-by-step explanation:
given :
- If (2,4),(2,6) are two vertices of an equilateral triangle then the third vertex is
to find :
- triangle then the third vertex is
solution :
- answer in attachment please check
answer :
- x= 2+5
