Question

pythagorean distance
what is the horizontal distance a?
pythagorean 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:

\(4\)