Question

find the distance between the two points in simplest radical form.

Explanation:

Step1: Identify coordinates

Top point: \((-5, 9)\), bottom point: \((4, -3)\)

Step2: Apply distance formula

Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Substitute \(x_1 = -5\), \(y_1 = 9\), \(x_2 = 4\), \(y_2 = -3\)
\(d = \sqrt{(4 - (-5))^2 + (-3 - 9)^2}\)

Step3: Simplify inside radicals

\(4 - (-5) = 9\), \(-3 - 9 = -12\)
\(d = \sqrt{9^2 + (-12)^2} = \sqrt{81 + 144}\)

Step4: Add and simplify radical

\(81 + 144 = 225\), \(\sqrt{225} = 15\)? Wait, no: Wait, \(9^2=81\), \((-12)^2=144\), \(81 + 144 = 225\)? Wait, no, 81+144=225? Wait, 81+144=225? Wait, 99=81, 1212=144, 81+144=225. Wait, but \(\sqrt{225}=15\). Wait, but let me check coordinates again. Wait, top point: looking at the graph, x=-5, y=9? Wait, the y-axis: top point is at y=9? Wait, the grid: each square is 1 unit. Top point: x=-5, y=9? Bottom point: x=4, y=-3. So horizontal difference: 4 - (-5) = 9, vertical difference: -3 - 9 = -12. Then squared: 81 + 144 = 225, square root is 15. Wait, but maybe I misread coordinates. Wait, top point: x=-5, y=9? Wait, the y-axis: the top point is at y=9? Let me check the graph again. The top yellow dot: x=-5, y=9? The bottom yellow dot: x=4, y=-3. So yes. Then distance is 15? Wait, but let's recalculate. \( (4 - (-5)) = 9 \), \( (-3 - 9) = -12 \). Then \( 9^2 + (-12)^2 = 81 + 144 = 225 \), so \( \sqrt{225} = 15 \). Wait, but maybe the top point is y=9? Wait, the y-axis: from 0 to 10, so 9 is correct. Bottom point: y=-3, correct. So distance is 15. Wait, but let me confirm coordinates again. Top point: x=-5, y=9. Bottom point: x=4, y=-3. So horizontal change: 4 - (-5) = 9, vertical change: -3 - 9 = -12. Then distance is \(\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15\). Yes.

Answer:

15