State whether the following statement is true or false. Justify your answer.
The points $(4,5), (7,6)$ and $(6,3)$ are collinear.
The points $(4,5), (7,6)$ and $(6,3)$ are collinear.
(B) False.
To check if the points $A(4,5), B(7,6)$ and $C(6,3)$ are collinear,we calculate the area of the triangle formed by these points using the formula:
Area $= \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
Substituting the values:
Area $= \frac{1}{2} |4(6 - 3) + 7(3 - 5) + 6(5 - 6)|$
Area $= \frac{1}{2} |4(3) + 7(-2) + 6(-1)|$
Area $= \frac{1}{2} |12 - 14 - 6|$
Area $= \frac{1}{2} |-8| = 4 \text{ sq. units}$.
Since the area of the triangle is not $0$,the points are not collinear.
To check if the points $A(4,5), B(7,6)$ and $C(6,3)$ are collinear,we calculate the area of the triangle formed by these points using the formula:
Area $= \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
Substituting the values:
Area $= \frac{1}{2} |4(6 - 3) + 7(3 - 5) + 6(5 - 6)|$
Area $= \frac{1}{2} |4(3) + 7(-2) + 6(-1)|$
Area $= \frac{1}{2} |12 - 14 - 6|$
Area $= \frac{1}{2} |-8| = 4 \text{ sq. units}$.
Since the area of the triangle is not $0$,the points are not collinear.
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