To determine the possible lengths of the diagonals of a rhombus given that each side measures 25, we can use the properties of a rhombus. In a rhombus, the diagonals bisect each other at right angles. Therefore, let's denote the lengths of the diagonals as \(d_1\) and \(d_2\).
From the properties of the rhombus, we can derive that the half-diagonals form right triangles with the sides. Each half-length of the diagonals is given by \( \frac{d_1}{2} \) and \( \frac{d_2}{2} \). According to the Pythagorean theorem:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 25^2 \]
This simplifies to:
\[ \frac{d_1^2}{4} + \frac{d_2^2}{4} = 625 \]
Multiplying through by 4, we get:
\[ d_1^2 + d_2^2 = 2500 \]
Now, we can evaluate each option provided:
A. \(d_1 = 22, d_2 = 40\)
\[ 22^2 + 40^2 = 484 + 1600 = 2084 \quad (\text{Not equal to } 2500) \]
B. \(d_1 = 26, d_2 = 36\)
\[ 26^2 + 36^2 = 676 + 1296 = 1972 \quad (\text{Not equal to } 2500) \]
C. \(d_1 = 26, d_2 = 48\)
\[ 26^2 + 48^2 = 676 + 2304 = 2980 \quad (\text{Not equal to } 2500) \]
D. \(d_1 = 30, d_2 = 40\)
\[ 30^2 + 40^2 = 900 + 1600 = 2500 \quad (\text{Equal to } 2500) \]
Only option D satisfies the equation \(d_1^2 + d_2^2 = 2500\). Thus, the lengths of the diagonals could be:
D. 30 and 40.