Question

pythagorean distance
what is the vertical distance b?
pythagorean formula \\( c^2 = a^2 + b^2 \\)
\\( a = 8 \\)
\\( b = ? \\)

Explanation:

Step1: Identify coordinates of points

The upper point is at \((5, 3)\) and the lower point is at \((-3, -3)\). Vertical distance \(b\) is the difference in \(y\)-coordinates.

Step2: Calculate vertical difference

\(b = 3 - (-3) = 6\)? Wait, no, wait. Wait, the horizontal distance \(a = 8\) (from \(x=-3\) to \(x = 5\), \(5 - (-3)=8\)). Wait, maybe using Pythagorean theorem. Wait, the upper point's \(y\) is 3, lower point's \(y\) is -3. So vertical distance is \(3 - (-3)=6\)? Wait, but let's check the grid. Wait, the upper blue dot is at \(y = 3\), lower blue dot (the vertical dashed line) is at \(y=-3\). So vertical distance \(b = 3 - (-3)=6\)? Wait, but let's check the Pythagorean formula. Wait, maybe I misread. Wait, the horizontal distance \(a = 8\) (from \(x=-3\) to \(x = 5\), \(5 - (-3)=8\)). The vertical distance: upper point \(y=3\), lower point \(y=-3\), so \(b = 3 - (-3)=6\). Wait, but let's confirm. Alternatively, maybe the two points are \((5, 3)\) and \((-3, -3)\). So vertical change is \(3 - (-3)=6\), horizontal change is \(5 - (-3)=8\). Then \(c^2=8^2 + 6^2=64 + 36 = 100\), \(c = 10\). But the question is vertical distance \(b\), which is the difference in \(y\)-coordinates. So \(b = 3 - (-3)=6\). Wait, but let's check the grid. The upper point is at \(x = 5\), \(y = 3\); the lower point (the blue dot on the vertical dashed line) is at \(x = 5\), \(y=-3\). So the vertical distance between \((5, 3)\) and \((5, -3)\) is \(3 - (-3)=6\). So \(b = 6\).

Wait, but the options have 6? Wait, the right panel has 6 as an option (the first row of numbers, 6 is there). Wait, maybe I made a mistake earlier. Wait, the vertical distance is the difference in \(y\)-values. So \(3 - (-3)=6\), so \(b = 6\).

Answer:

6