Question

the coordinates of the vertices of a right triangle are a(-5, 2), b(-5, -4), and c(5, -4). which one of the following measurements is closest to the length of segment ac? a 12.8 units b 10.6 units c 12.5 units d 11.7 units

Explanation:

Step1: Identify the coordinates of A and C

A is \((-5, 2)\) and C is \((5, -4)\).

Step2: Use the distance formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
Substitute \(x_1 = -5\), \(y_1 = 2\), \(x_2 = 5\), \(y_2 = -4\) into the formula:
\[

$$\begin{align*} d&=\sqrt{(5 - (-5))^2 + (-4 - 2)^2}\\ &=\sqrt{(10)^2 + (-6)^2}\\ &=\sqrt{100 + 36}\\ &=\sqrt{136} \end{align*}$$

\]

Step3: Calculate the approximate value of \(\sqrt{136}\)

\(\sqrt{121} = 11\), \(\sqrt{144} = 12\), so \(\sqrt{136}\) is between 11 and 12.
\(\sqrt{136} \approx 11.66\), which is closest to 11.7.

Answer:

D. 11.7 units