Question

use this explore tool to investigate the distance formula. to use the tool, press on the coordinate plane to move point c. recall that the distance formula is $sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=$distance. to determine the distance between points d and l, start by dragging point c to form right triangle cdl so that the legs are parallel to the axes. there are two locations for point c that will form a right triangle: (-4, 10) and (11, -9). in both places where you can place point c to form the right triangle cdl, the horizontal distance between points d and l, $|x_2 - x_1|=$ blocks, and the vertical distance between points d and l, $|y_2 - y_1|=$ blocks. use the distance formula to determine the distance between points d and l. $(x_2 - x_1)^2=$ enter the square of the horizontal distance. $(y_2 - y_1)^2=$ enter the square of the vertical distance. $(x_2 - x_1)^2+(y_2 - y_1)^2=$ enter the sum of the squares of the horizontal and vertical distances. $sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}approx$ enter the approximate square root of the sum.

Explanation:

Step1: Find horizontal distance

Let's assume the coordinates of \(D\) and \(L\) such that for the horizontal distance, if the \(x -\)coordinates of \(D\) and \(L\) are \(x_D\) and \(x_L\). Given the two possible \(C\) - points \((-4,10)\) and \((11, - 9)\), we can find the horizontal distance. If we consider the \(x\) - values, assume \(x_D = 11\) and \(x_L=-4\), then \(|x_2 - x_1|=|11-(-4)| = 15\) blocks.

Step2: Find vertical distance

For the vertical distance, if the \(y\) - coordinates of \(D\) and \(L\) are \(y_D\) and \(y_L\). Assume \(y_D = 10\) and \(y_L=-9\), then \(|y_2 - y_1|=|10 - (-9)|=19\) blocks.

Step3: Calculate square of horizontal distance

\((x_2 - x_1)^2=15^2 = 225\)

Step4: Calculate square of vertical distance

\((y_2 - y_1)^2=19^2=361\)

Step5: Calculate sum of squares

\((x_2 - x_1)^2+(y_2 - y_1)^2=225 + 361=586\)

Step6: Calculate square - root

\(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}=\sqrt{586}\approx24.2\)

Answer:

Horizontal distance: 15, Vertical distance: 19, \((x_2 - x_1)^2:225\), \((y_2 - y_1)^2:361\), \((x_2 - x_1)^2+(y_2 - y_1)^2:586\), \(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\approx24.2\)