If the points (3, -2), (x, 2), (8, 8) are collinear, then find the value of x.
Answers
Step-by-step explanation:
Given :-
The points (3, -2), (x, 2), (8, 8) are collinear.
To find :-
Find the value of x ?
Solution :-
(3,-2)________(x,2)_______(8,8)
Given points are :(3, -2), (x, 2), (8, 8)
Let (x1, y1)=(3,-2)=> x1 = 3 and y1 = -2
Let (x2, y2)=(x,2)=> x2 = x and y2 = 2
Let (x3, y3)=(8,8)=>x3 = 8 and y3 = 8
We know that
If the points A, B, C are collinear then the area of the triangle formed by the points is equal to Zero
Area of a triangle =
∆=(1/2) | x1(y2-y3)+x2(y3-y1)+x3(y1-y2) | sq units
So we have
Given points are collinear points.
Area of the triangle = 0
=> ∆=(1/2) | x1(y2-y3)+x2(y3-y1)+x3(y1-y2) | = 0
On Substituting these values in the above formula then
=> (1/2) | 3(2-8)+x(8-(-2))+8(-2-2) | = 0
=> (1/2) | 3(-6)+x(8+2)+8(-4) | = 0
=> (1/2) | (-18)+x(10)+(-32) | = 0
=> (1/2) | -18+10x-32 |= 0
=> (1/2) | 10x-50 | = 0
=> (1/2) (10x-50) = 0
=> (1/2)(2)(5x-25) = 0
=>(2/2)(5x-25) = 0
=> 5x-25 = 0
=> 5x = 25
=> x = 25/5
=> x = 5
Therefore, x = 5
Answer:-
The value of x for the given problem is 5
Check:-
If x = 5 then points are (3, -2), (5, 2), (8, 8)
Area of the triangle
=> (1/2) | 3(2-8)+5(8-(-2))+8(-2-2) |
=> (1/2) | 3(-6)+5(10)+8(-4) |
=> (1/2) | -18+50-32 |
=> (1/2) | 50-50 |
=> (1/2) | 0 |
=> 0/2
=> 0 sq.units
Since Area is zero the given points are Collinear .
Verified the given relations in the given problem.
Used Concept :-
If the points A, B, C are collinear then the area of the triangle formed by the points is equal to Zero
Used formulae:-
Area of a triangle is formed by the given points
∆=(1/2) | x1(y2-y3)+x2(y3-y1)+x3(y1-y2) | sq units
Points to know :-
- The points lie on the same line are called Collinear points.
- The points A,B and C are collinear then AB+BC= AC.
Answer:
x=11
Step-by-step explanation:
Let A=(3,-2) ,B=(x,2) and C=(8,8)
AB²=(3-x)²+(-2-2)²
=(3)²+(x)²-2×3x+(-4)²
=9+x²-6x+16
=x²-6x+25
AB=√x²-6x+25
BC²=(x-8)²+(2-8)²
=(x)²+(8)²-2×8x+(-6)²
=x²+64-16x+36
=x²-16x+100
BC=√x²-16x+100
AC²=(8-3)²+(8+2)²
=(5)²+(10)²
=25+100
=125
AC=√125
AB+BC=AC
=>√x²-6x+25+√x²-16x+100=√125
Squaring both sides:-
=>x²-6x+25+x²-16x+100=125
=>2x²-22x+125=125
=>2x²=22x
=>x²=11x
=>x=11
And that's all your answer