Question

midpoint and distance
problem
what is the distance between ( a(-2,3) ) and ( b(4,14) )? round to the nearest tenth.
justify your steps.
let ( a(-2,3) ) be ( (x_1,y_1) ) and ( b(4,14) ) be ( (x_2,y_2) ).
( d=sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} )
write the distance formula.
( d=sqrt{(4-(-2))^2+(14 - 3)^2} )
substitute ( x_1,x_2,y_1 ), and ( y_2 ) into the formula,
( d=sqrt{6^2+11^2} )
subtract.
( d=sqrt{36 + 121} )
square each term.
( d=sqrt{157} )
add.
( d = 12.53 )
use a calculator to find the square root.
( d = 12.5 )
round to the nearest tenth.

Explanation:

Step1: Substitute coordinates into distance formula

Given \(A(-2,3)\) as \((x_1,y_1)\) and \(B(4,14)\) as \((x_2,y_2)\), use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Substitute \(x_1=-2,x_2 = 4,y_1 = 3,y_2=14\) into the formula:
\(d=\sqrt{(4-(-2))^2+(14 - 3)^2}=\sqrt{(4 + 2)^2+(11)^2}\)

Step2: Calculate the squares

\((4 + 2)^2=6^2 = 36\) and \((11)^2=121\).
So \(d=\sqrt{36+121}\)

Step3: Add and find the square - root

\(36+121 = 157\), then \(d=\sqrt{157}\approx12.53\)

Step4: Round to the nearest tenth

Rounding \(12.53\) to the nearest tenth gives \(d = 12.5\)

Answer:

The distance between \(A(-2,3)\) and \(B(4,14)\) is \(12.5\)