Question
A rhombus has side lengths of 25. What could be the lengths of the diagonals?
A. 22 and 40
B. 26 and 36
C. 26 and 48
D. 30 and 40
A. 22 and 40
B. 26 and 36
C. 26 and 48
D. 30 and 40
Answers
GPT-4o mini
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.**
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.**