Question

1. the length of \\(\overline{fh}\\) is 9 cm. what is the length of \\(\overline{gk}\\)?
2. what is the area of rectangle fghk?

Explanation:

Step1: Recall rectangle diagonal property

In a rectangle, diagonals are congruent (equal in length).

Step2: Apply property to find $\overline{GK}$

Given $\overline{FH}=9$ cm, so $\overline{GK}=\overline{FH}$.

Step1: Find length of the rectangle

Use Pythagorean theorem on $\triangle FGK$. Let $FK = l$, $FG=4$ cm, $GK=9$ cm.
$$l = \sqrt{GK^2 - FG^2} = \sqrt{9^2 - 4^2}$$
$$l = \sqrt{81 - 16} = \sqrt{65} \text{ cm}$$

Step2: Calculate rectangle area

Area = length $\times$ width = $l \times FG$
$$\text{Area} = 4 \times \sqrt{65}$$

Answer:

9 cm

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