Question

pythagorean distance
what is the horizontal distance a?
(there is a coordinate grid with points, and the pythagorean formula c² = a² + b² at the bottom, and a number pad on the right with numbers and square roots.)

Explanation:

Step1: Identify coordinates of points

The blue points: left at (2,1), right at (4,2). So vertical distance \( b = 2 - 1 = 1 \), hypotenuse \( c \) (but we can find \( a \) directly by horizontal difference? Wait, no, wait: horizontal distance \( a \) is \( 4 - 2 = 2 \)? Wait, no, maybe using Pythagorean. Wait, the horizontal distance between x=2 and x=4 is \( 4 - 2 = 2 \)? Wait, no, the right point is at x=4, y=2; left at x=2, y=1. So horizontal change (a) is \( 4 - 2 = 2 \)? Wait, no, the vertical change (b) is \( 2 - 1 = 1 \). Wait, maybe I misread. Wait, the dashed line is vertical, so \( b \) is vertical distance: from y=1 to y=2, so \( b = 1 \). The horizontal distance \( a \) is from x=2 to x=4, so \( a = 4 - 2 = 2 \)? Wait, but the options have 2? Wait, the right panel has numbers, 0,1,2,... Wait, the left point is at x=2, y=1; right at x=4, y=2. So horizontal distance (a) is \( 4 - 2 = 2 \). Wait, but let's check with Pythagorean. Wait, maybe the hypotenuse length? No, the question is horizontal distance \( a \), which is the difference in x-coordinates. So \( a = 4 - 2 = 2 \). Wait, but the options: looking at the right panel, the top row has 0,1,2. Wait, the second column, first row? Wait, no, the right panel's first row: 0,1,2. So 2 is an option (first row, third column: 2). Wait, maybe I made a mistake. Wait, the horizontal distance between x=2 and x=4 is 2 units (since each grid is 1 unit). So \( a = 2 \).

Wait, let's confirm with Pythagorean. Let's find \( c \) first. The two points: (2,1) and (4,2). The distance between them (c) is \( \sqrt{(4-2)^2 + (2-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.2 \). Then using \( c^2 = a^2 + b^2 \), \( (\sqrt{5})^2 = a^2 + 1^2 \), \( 5 = a^2 + 1 \), so \( a^2 = 4 \), \( a = 2 \). Yes, that matches. So \( a = 2 \).

Step2: Find \( a \) from x-coordinates

Horizontal distance is difference in x: \( 4 - 2 = 2 \).

Answer:

2 (from the right panel, first row, third column: 2)