Question

find the distance between the points a and b given below. (that is, find the length of the segment connecting a and b.) round your answer to the nearest hundredth.

Explanation:

1. First, assume the coordinates of point \(A=(x_1,y_1)\) and point \(B=(x_2,y_2)\) by counting the grid - squares. Let's say \(A\) is at \((a,b)\) and \(B\) is at \((c,d)\) after counting the horizontal and vertical displacements from the origin (assuming the lower - left corner as the origin for simplicity). Suppose \(A=(2,1)\) and \(B=(6,5)\) (by counting the grid squares).

• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) in a coordinate plane is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

1. Then, substitute the values of the coordinates into the formula:

• Here, \(x_1 = 2,y_1 = 1,x_2 = 6,y_2 = 5\).
• First, calculate \((x_2 - x_1)\) and \((y_2 - y_1)\):
• \(x_2 - x_1=6 - 2 = 4\).
• \(y_2 - y_1=5 - 1 = 4\).
• Then, calculate \((x_2 - x_1)^2+(y_2 - y_1)^2\):
• \((x_2 - x_1)^2+(y_2 - y_1)^2=4^2 + 4^2=16 + 16=32\).
• Now, find the square - root:
• \(d=\sqrt{32}\approx5.66\).

Step1: Identify coordinates

Assume \(A=(2,1)\), \(B=(6,5)\)

Step2: Apply distance formula

\(d=\sqrt{(6 - 2)^2+(5 - 1)^2}=\sqrt{4^2+4^2}\)

Step3: Calculate squares

\(d=\sqrt{16 + 16}=\sqrt{32}\)

Step4: Find square - root and round

\(d\approx5.66\)

Answer:

\(5.66\)