Question

1. without using a coordinate plane, what is the distance between (2,11) and (12,3)

Explanation:

Step1: Identify the 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, \((x_1,y_1)=(2,11)\) and \((x_2,y_2)=(12,3)\).

Step2: Calculate the differences

First, find \( x_2 - x_1 = 12 - 2 = 10 \) and \( y_2 - y_1 = 3 - 11 = - 8 \).

Step3: Square the differences

Square the results: \( (x_2 - x_1)^2 = 10^2 = 100 \) and \( (y_2 - y_1)^2 = (-8)^2 = 64 \).

Step4: Sum the squares

Add the squared values: \( 100 + 64 = 164 \).

Step5: Take the square root

Take the square root of the sum: \( d=\sqrt{164}=\sqrt{4\times41}=2\sqrt{41}\approx12.806 \) (if a decimal approximation is needed) or keep it as \( 2\sqrt{41} \). But if we consider the problem might have a typo (maybe points are \((2,11)\) and \((12,3)\) or maybe \((2,11)\) and \((12,23)\)? Wait, the original points: let's re - check. If the points are \((2,11)\) and \((12,3)\):

Wait, maybe I misread the second point. Let's assume the second point is \((12,3)\) (as per the image). Then:

Wait, maybe the user made a typo, but following the given points \((2,11)\) and \((12,3)\):

Wait, let's recalculate:

\( x_1 = 2,y_1 = 11,x_2 = 12,y_2 = 3 \)

\( \Delta x=12 - 2 = 10 \), \( \Delta y=3 - 11=-8 \)

\( d=\sqrt{10^{2}+(-8)^{2}}=\sqrt{100 + 64}=\sqrt{164}=2\sqrt{41}\approx12.81 \)

But if the second point is \((12,23)\) (maybe a typo in the image), then \( \Delta y = 23 - 11 = 12 \), \( \Delta x=12 - 2 = 10 \), \( d=\sqrt{10^{2}+12^{2}}=\sqrt{100 + 144}=\sqrt{244}=\sqrt{4\times61}=2\sqrt{61}\approx15.62 \)

But based on the given image: the points are \((2,11)\) and \((12,3)\) (assuming the last digit of the second point is 3). So the distance is \( \sqrt{(12 - 2)^2+(3 - 11)^2}=\sqrt{100 + 64}=\sqrt{164}=2\sqrt{41}\approx12.81 \)

Answer:

If the points are \((2,11)\) and \((12,3)\), the distance is \( 2\sqrt{41}\) (or approximately \( 12.81\)). If there is a typo and the second point is \((12,23)\), the distance is \( 2\sqrt{61}\) (or approximately \( 15.62\)). But based on the given text in the image, with points \((2,11)\) and \((12,3)\), the distance is \( \sqrt{164}=2\sqrt{41}\approx12.81\)