Question

pythagorean distance
what is the vertical distance b?
pythagorean formula \\( c^2 = a^2 + b^2 \\)
(the image contains a coordinate grid with two blue points, a line segment c, horizontal segment a=4, vertical segment b=?, and a number pad on the right with numbers and square roots.)

Explanation:

Step1: Identify y - coordinates

The upper point has y - coordinate 4, the lower point has y - coordinate 3.

Step2: Calculate vertical distance

Vertical distance \( b=\vert4 - 3\vert= 1\)? Wait, no, wait. Wait, the Pythagorean formula is \( c^{2}=a^{2}+b^{2}\). Wait, first, we need to find the length of \( c\)? No, wait, the vertical distance is the difference in y - values. Wait, the upper blue dot is at y = 4, the lower blue dot (the one with the dashed line) is at y = 3? Wait, no, looking at the graph, the upper point is at (2,4) and the lower point (the one connected by the vertical dashed line) is at (2,3). So the vertical distance \( b = 4-3=1\)? But that doesn't match the options. Wait, maybe I misread. Wait, the horizontal distance \( a = 4\). Wait, let's check the coordinates. The left blue dot: let's find its coordinates. The left blue dot is at, say, x=-2, y = 3 (since it's on the dashed horizontal line at y = 3). The right blue dot is at x = 2, y = 4. So horizontal distance \( a=\vert2-(-2)\vert = 4\) (correct, as given \( a = 4\)). Vertical distance \( b=\vert4 - 3\vert=1\)? But 1 is not in the options. Wait, maybe the Pythagorean theorem is used here. Wait, \( c\) is the length of the hypotenuse (the black line). Let's find \( c\) first? Wait, no, the question is vertical distance \( b\). Wait, maybe the y - coordinates are different. Wait, the upper dot is at y = 4, the lower dot (the one with the vertical dashed line) is at y = 3? Wait, no, maybe the left dot is at (x1,y1) and the right dot is at (x2,y2). Let's get the coordinates: left dot: let's see, the horizontal dashed line is at y = 3, so left dot is (x1, 3), right dot is (x2, 4). The horizontal distance \( a=x2 - x1=4\) (since \( a = 4\)). Now, vertical distance \( b = 4 - 3=1\)? But 1 is not in the options. Wait, maybe I made a mistake. Wait, the options have 1? No, the options on the right: 0,1,2,3,4,5,6,7,8,9,10,12,13,14,15, and radicals. Wait, maybe the vertical distance is calculated as the difference in y - values, but maybe the left dot is at (x1,y1) and the right dot is at (x2,y2), so \( b=\vert y2 - y1\vert\). Wait, left dot: let's find x - coordinate. The horizontal distance \( a = 4\), so if right dot is at x = 2, then left dot is at x=2 - 4=-2. So left dot is (-2, 3), right dot is (2, 4). So vertical distance \( b = 4 - 3 = 1\). But 1 is an option (the second row, first column? Wait, the options on the right: first row 0,1,2; second row 3,4,5; third row 6,7,8; fourth row 9,10,12; fifth row 13,14,15; sixth row radicals; seventh row radicals; eighth row radicals. Wait, 1 is in the first row, second column (0,1,2: 1 is the middle one? Wait, the first row: 0 (first), 1 (second), 2 (third). So 1 is an option. Wait, maybe I was overcomplicating. The vertical distance is the difference in y - coordinates. The upper point is at y = 4, the lower point (the one with the vertical dashed line) is at y = 3, so \( b=4 - 3 = 1\). But wait, the Pythagorean formula is given, but maybe the question is using the Pythagorean theorem to find \( b\) if we know \( a\) and \( c\)? Wait, no, the question is "What is the vertical distance b?". Vertical distance is just the difference in y - values. So the two points have y - coordinates 3 and 4, so the vertical distance is \( 4 - 3=1\). But 1 is an option (the second button in the first row of the number pad: 0,1,2; 1 is the middle one). Wait, maybe I misread the y - coordinates. Wait, the upper dot is at y = 4, the lower dot (the one with the vertical dashed line) is at y = 3, so vertical distance is 1. But let's check again. The left dot is on the horizontal line y = 3, the right dot is on y = 4. So vertical distance \( b = 4 - 3 = 1\). So the answer should be 1? But 1 is in the options (the second number in the first row: 0,1,2; 1 is the middle one). Wait, maybe the problem is that I thought the vertical distance is 1, but let's confirm. The vertical distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(\vert y_2 - y_1\vert\). Here, \( y_1 = 3\), \( y_2 = 4\), so \( b=\vert4 - 3\vert = 1\). So the answer is 1. But looking at the number pad, 1 is the second button in the first row (0,1,2). Wait, the first row: 0 (first), 1 (second), 2 (third). So the answer is 1.
Wait, no, maybe I made a mistake. Wait, the horizontal distance \( a = 4\), and if we use the Pythagorean theorem, maybe \( c\) is known? Wait, no, the question is vertical distance, which is just the difference in y - coordinates. So the vertical distance is 1. So the answer is 1. But let's check the options again. The number pad has 0,1,2 in the first row. So 1 is an option. So the vertical distance \( b = 1\).

Wait, but maybe the coordinates are different. Let's re - examine the graph. The left blue dot: let's find its x - coordinate. The horizontal dashed line is at y = 3, and the horizontal distance \( a = 4\), so the right blue dot is at x = 2 (since from x=-2 to x = 2 is 4 units). The right blue dot is at (2,4), the left blue dot is at (-2,3). So vertical distance is \( 4 - 3 = 1\). So \( b = 1\).

Answer:

1 (from the number pad, the second button in the first row: 0,1,2)