Question

what is the area of rectangle rstu? figure 1 shows a rectangle rstu with diagonals intersecting at point v. the length of ru is 4, and the length of tv (or part of the diagonal) is 5.

Explanation:

Step1: Find diagonal length of rectangle

In a rectangle, diagonals are equal and bisect each other. Given half-diagonal $VU=5$, so full diagonal $SU=2\times5=10$.

Step2: Calculate height of rectangle

Use Pythagorean theorem on $\triangle RSU$: Let height $SR = h$, base $RU=4$, diagonal $SU=10$.
$$h = \sqrt{SU^2 - RU^2} = \sqrt{10^2 - 4^2}$$
$$h = \sqrt{100 - 16} = \sqrt{84} = 2\sqrt{21}$$

Step3: Compute area of rectangle

Area = length $\times$ width = $RU \times SR$
$$\text{Area} = 4 \times 2\sqrt{21}$$

Answer:

$8\sqrt{21}$ (or approximately 36.66 if a decimal is needed)