Points A (–6, 10), B (–4, 6) and C (3, –8) are collinear such that AB = 2/9 AC. Is the following statement true or false
Solution:
Given, the points are A(-6, 10) B(-4, 6) and C(3, -8)
We have to check if the points are collinear and prove that AB = 2/9 AC.
The area of a triangle with vertices A (x₁ , y₁) , B (x₂ , y₂) and C (x₃ , y₃) is
1/2[x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)]
To check for the points to be collinear, the area of the triangle must be zero.
Here, (x₁ , y₁) = (-6, 10), (x₂ , y₂) = (-4, 6) and (x₃ , y₃) = (3, -8)
Area of triangle = 1/2[-6(6 - (-8)) + -4(-8 - 10) + 3(10 - 6)
= 1/2[-6(14) + -4(-18) + 3(4)]
= 1/2[-84 + 72 + 12]
= 1/2[-84 + 84]
= 0
Area of the triangle = 0
Therefore, the points are collinear.
The distance between two points P (x₁ , y₁) and Q (x₂ , y₂) is
√[(x₂ - x₁)² + (y₂ - y₁)²]
The distance between the points A(-6, 10) and B(-4, 6)
AB = √[(-4 - (-6))² + (6 - 10)²]
= √[(2)² + (-4)²]
= √(4 + 16)
= √20
= 2√5
The distance between the points A(-6, 10) and C(3, -8)
AC = √[(3 - (-6))² + (-8 - 10)²]
= √[(9)² + (-18)²]
= √(81 + 324)
= √405
= 9√5
Now, 2/9 AC = 2/9(9√5) = 2√5 = AB
Therefore, AB = 2/9 AC
✦ Try This: Find the value of m, if the points (5, 1), (-2, -3) and (8, 2m) are collinear.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7
NCERT Exemplar Class 10 Maths Exercise 7.2 Problem 10
Points A (–6, 10), B (–4, 6) and C (3, –8) are collinear such that AB = 2/9 AC. Is the following statement true or false
Summary:
The statement “Points A (–6, 10), B (–4, 6) and C (3, –8) are collinear such that AB = 2/9 AC” is true as it satisfies the condition of collinearity i.e., area of the triangle is equal to zero
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