Question

find the distance between the pair of points. also, find the midpoint of the line segment joining them. (7,4), (7 + 6√3,10) find the distance. d = □

Explanation:

Step1: Recall distance formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \( x_1 = 7 \), \( y_1 = 4 \), \( x_2 = 7 + 6\sqrt{3} \), \( y_2 = 10 \).

Step2: Calculate \( x_2 - x_1 \) and \( y_2 - y_1 \)

\( x_2 - x_1=(7 + 6\sqrt{3})-7 = 6\sqrt{3} \)
\( y_2 - y_1 = 10 - 4 = 6 \)

Step3: Substitute into distance formula

\( d=\sqrt{(6\sqrt{3})^2 + 6^2} \)

Step4: Simplify the expression

First, calculate \((6\sqrt{3})^2 = 6^2\times(\sqrt{3})^2 = 36\times3 = 108\) and \( 6^2 = 36 \). Then, \( d=\sqrt{108 + 36}=\sqrt{144} \)

Step5: Evaluate the square root

\( \sqrt{144}=12 \)

Answer:

\( 12 \)