Question

pythagorean distance
what is the horizontal distance a?
pythagorean formula c² = a² + b²

Explanation:

Step1: Identify coordinates of points

Left point: Let's say its x - coordinate is - 3, right point (for horizontal distance) has x - coordinate 5. Wait, actually, the horizontal distance \(a\) is the difference in x - coordinates. Wait, looking at the graph, the left blue point is at \(x=-3\) (since it's 3 units left of the origin on x - axis) and the right blue point (for horizontal line) is at \(x = 5\). So the horizontal distance \(a\) is \(|5-(-3)|\).
\(a=5 - (-3)=8\)? Wait, no, wait the vertical distance \(b\): the top blue point is at \(y = 3\), bottom blue point (for vertical line) is at \(y=-3\), so \(b=|3-(-3)| = 6\). Then the length of the hypotenuse \(c\): let's check the coordinates of the two end - points of the line segment. Left end - point: \((-3,-3)\), right end - point: \((5,3)\). So the horizontal change (which is \(a\)) is \(5-(-3)=8\), vertical change (which is \(b\)) is \(3 - (-3)=6\). Let's verify with Pythagorean theorem. \(c^{2}=a^{2}+b^{2}\), \(a = 8\), \(b = 6\), so \(c^{2}=64 + 36=100\), \(c = 10\). Wait, but let's check the x - coordinates for the horizontal distance. The left point (the one at the bottom left) is at \(x=-3\), and the point directly below the top right point (the blue dot on the vertical line) is at \(x = 5\). So the horizontal distance \(a\) is \(5-(-3)=8\).

Step2: Confirm with coordinates

The two points that define the horizontal distance are \((-3,-3)\) and \((5,-3)\) (since the horizontal line is at \(y=-3\)). The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\). For horizontal distance, \(y_1=y_2=-3\), so the distance \(a=\sqrt{(5 - (-3))^{2}+(-3-(-3))^{2}}=\sqrt{(8)^{2}+0^{2}} = 8\).

Answer:

8