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Question: What is the distance between the points \( G\left( {10, - 8} \right) \) and \( H\left( { - 3, - 2} \...

What is the distance between the points G(10,−8)G\left( {10, - 8} \right) and H(−3,−2)H\left( { - 3, - 2} \right) ?
(A) 205\sqrt {205}
(B) 315\sqrt {315}
(C) 305\sqrt {305}
(D) 215\sqrt {215}

Explanation

Solution

Hint : In the given question, we are required to find the distance between the two given points. We will use the distance formula, that is,
d=(x2−x1)2+(y2−y1)2d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}
Where, x2={x_2} = x-coordinate of second point
x1={x_1} = x-coordinate of first point
y2={y_2} = y-coordinate of second point
y1={y_1} = y-coordinate of first point
We must know how to calculate square and square roots of numbers to calculate the distance between the two given points and match the options.

Complete step-by-step answer :
The two given points are: G(10,−8)G\left( {10, - 8} \right) and H(−3,−2)H\left( { - 3, - 2} \right) .
Now, by using the distance formula between two points, i.e., d=(x2−x1)2+(y2−y1)2d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} , we can calculate the distance between the points G(10,−8)G\left( {10, - 8} \right) and H(−3,−2)H\left( { - 3, - 2} \right) .
Now, considering the point G(10,−8)G\left( {10, - 8} \right) as the first point, we get x1=10{x_1} = 10 and y1=−8{y_1} = - 8 .
Similarly, considering the point H(−3,−2)H\left( { - 3, - 2} \right) as second, we get, x2=−3{x_2} = - 3 and y2=−2{y_2} = - 2 .
Now, substituting the values of known entities, we get,
GH=(−3−10)2+(−2−(−8))2GH = \sqrt {{{\left( { - 3 - 10} \right)}^2} + {{\left( { - 2 - \left( { - 8} \right)} \right)}^2}}
Simplifying the expression by opening the brackets, we get,
⇒GH=(−13)2+(6)2\Rightarrow GH = \sqrt {{{\left( { - 13} \right)}^2} + {{\left( 6 \right)}^2}}
Since we know that the square of 1313 is 169169 and the square of 66 is 3636 . So, we get,
⇒GH=169+36\Rightarrow GH = \sqrt {169 + 36}
Adding up like terms,
⇒GH=205\Rightarrow GH = \sqrt {205}
So, the distance between the points G(10,−8)G\left( {10, - 8} \right) and H(−3,−2)H\left( { - 3, - 2} \right) is 205\sqrt {205} units.
Therefore, option (A) is the correct answer.
So, the correct answer is “Option A”.

Note : We must know the distance formula to calculate the length of a line segment when the endpoints are given or the distance between any two given points. Care must be taken while computing the squares and doing calculations so as to be sure of the final answer. The order in which the points are taken does not matter much as the squares of negative and positive entities having the same magnitude are equal.