Sean las diagonales de un rombo 24 cm y 10 cm. Halla el perímetro del rombo.
A rhombus is a quadrilateral with all four sides of equal length. A key property of a rhombus is that its diagonals bisect each other at right angles. These diagonals divide the rhombus into four congruent right-angled triangles. The legs of these right-angled triangles are exactly half the length of each diagonal. We are given the lengths of the diagonals as \(d_1 = 24 \text{ cm}\) and \(d_2 = 10 \text{ cm}\).
Each right-angled triangle formed by the intersection of the diagonals has legs of lengths:
\(a = \frac{d_1}{2} = \frac{24}{2} = 12 \text{ cm}\)
\(b = \frac{d_2}{2} = \frac{10}{2} = 5 \text{ cm}\)
The hypotenuse of these right-angled triangles corresponds to the side length \(s\) of the rhombus. Applying the Pythagorean theorem, \(s^2 = a^2 + b^2\):
\(s = \sqrt{12^2 + 5^2}\)
\(s = \sqrt{144 + 25}\)
\(s = \sqrt{169}\)
\(s = 13 \text{ cm}\)
The perimeter \(P\) of a rhombus with side length \(s\) is given by \(P = 4s\).
\(P = 4 \times 13 = 52 \text{ cm}\)
Comparing this result with the given options, we identify that the correct answer is 52 cm, which corresponds to option a.
Imagine you are crafting a diamond-shaped kite frame where the cross-bars (diagonals) measure 24 cm and 10 cm. Knowing the perimeter of the kite helps you determine the total length of the frame material needed to connect the four corners. In this case, you would need 52 cm of material to construct the perimeter of the kite.