Question

geometry - honors - 9(ac)10(a) - 2025 - 2026
topic 4: readiness assessment
use the figure shown.
what is the exact length of $overline{ac}$?
a 7.071
b 10
c $7sqrt{2}$
d $5sqrt{2}$

Explanation:

Step1: Count the horizontal and vertical displacements from A to C.

From the grid - based figure, moving from point A to point C, the horizontal displacement is 7 units and the vertical displacement is 7 units.

Step2: Apply the distance formula (derived from the Pythagorean theorem).

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). In a right - triangle formed by the horizontal and vertical displacements, if the horizontal side \(x = 7\) and the vertical side \(y = 7\), then the length of the hypotenuse \(AC\) (using the Pythagorean theorem \(a^2 + b^2=c^2\), where \(a = 7\), \(b = 7\) and \(c=AC\)) is \(AC=\sqrt{7^{2}+7^{2}}=\sqrt{49 + 49}=\sqrt{98}\).

Step3: Simplify the square - root expression.

\(\sqrt{98}=\sqrt{49\times2}=\sqrt{49}\times\sqrt{2}=7\sqrt{2}\).

Answer:

C. \(7\sqrt{2}\)