Question

How do you find the distance between the point A (-3, 5) and the point B (4,-6) in the coordinate plane?

Answer 1

Hint: This type of problem is based on the concept of geometry. We know that the distance between two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] in a coordinate plane is \[\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}\].
Here, we have been given the points are A (-3, 5) and B (4,-6). By comparing, we get \[{{x}_{1}}=-3\], \[{{y}_{1}}=5\], \[{{x}_{2}}=4\] and \[{{y}_{2}}=-6\]. Substitute these values in the formula to find the distance between A and B. Do necessary calculation and find the square root of the expression which is the required answer.

Complete step by step solution:
According to the question, we are asked to find the distance between A and B.
We have been given the points are A (-3, 5) and B (4,-6).
We know that the formula to find the distance between two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] is
\[d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}\] where d is the distance.
We know that the two pints are A (-3, 5) and B (4,-6).
By comparing with the known formula, we get
\[{{x}_{1}}=-3\], \[{{y}_{1}}=5\], \[{{x}_{2}}=4\] and \[{{y}_{2}}=-6\].
Let us now substitute the values in the distance formula.
\[\Rightarrow d=\sqrt{{{\left( 4-\left( -3 \right) \right)}^{2}}+{{\left( -6-5 \right)}^{2}}}\]
We know that \[-\left( -x \right)=x\]. Using this property, we get
\[\Rightarrow d=\sqrt{{{\left( 4+3 \right)}^{2}}+{{\left( -6-5 \right)}^{2}}}\]
On further calculations, we get
\[d=\sqrt{{{7}^{2}}+{{\left( -11 \right)}^{2}}}\]
We know that square of 7 is 49 and square of -11 is 121.
On substituting in d, we get
\[d=\sqrt{49+121}\]
We know that 49+121=170.
Therefore, we get
\[d=\sqrt{170}\]
We cannot further simplify the above expression.
Therefore, the distance between the point A (-3, 5) and the point B (4,-6) in the coordinate plane is \[\sqrt{170}\] units.

Note: Whenever you get this type of problems, we should always know the formula for finding the distance in a coordinate plane. We never get a negative number in the square root. We should not forget to put units in the final answer without which the answer is incomplete. Avoid calculation mistakes based on sign conventions.