pythagorean distance\nwhat is the horizontal distance a?\npythagorean formula \\( c^2 = a^2 + b^2 \\)

Explanation:

Step1: Identify coordinates of points

Left point: ((-2, 3)), right - upper point: ((2, 4)), right - lower point: ((2, 3)). The vertical distance (b) is the difference in (y) - coordinates: (b = 4 - 3=1). The length of the hypotenuse (c) can be found using the distance formula between ((-2, 3)) and ((2, 4)). The distance formula is (d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}), so (c=\sqrt{(2 - (-2))^2+(4 - 3)^2}=\sqrt{16 + 1}=\sqrt{17}\approx4.1) (or we can also use the Pythagorean theorem directly once we know (a) and (b)).

Step2: Apply Pythagorean theorem

We know the Pythagorean formula (c^{2}=a^{2}+b^{2}). We know (b = 1) and (c=\sqrt{17}) (or we can calculate (c) from the coordinates). Let's solve for (a): (a^{2}=c^{2}-b^{2}). Substituting (c^{2}=17) (since (c = \sqrt{17}), (c^{2}=17)) and (b^{2}=1) (since (b = 1)), we get (a^{2}=17 - 1=16). Then (a=\sqrt{16}=4). Alternatively, we can calculate the horizontal distance directly from the (x) - coordinates of the two points ((-2, 3)) and ((2, 3)). The horizontal distance (a) is (|2-(-2)|=4).

Answer:

(4)