A rhombus has four equal sides. The perimeter of the rhombus is 52 m. Perimeter = 4 side length. 52 m = 4 side length. Side length = 52 m / 4 = 13 m. The diagonals of a rhombus bisect each other at right angles. This property divides the rhombus into four congruent right-angled triangles. Let the diagonals be $d_1$ and $d_2$. The shorter diagonal is given as 10 m. Let $d_1 = 10$ m. The legs of each right-angled triangle are half the lengths of the diagonals, which are $d_1/2$ and $d_2/2$. The hypotenuse of each right-angled triangle is the side length of the rhombus. In this case, the lengths of the legs are $10/2 = 5$ m and $d_2/2$. The hypotenuse is 13 m. Using the Pythagorean theorem for one of the right-angled triangles: $(d_1/2)^2 + (d_2/2)^2 = (\text{side})^2$ $(5)^2 + (d_2/2)^2 = (13)^2$ $25 + (d_2/2)^2 = 169$ $(d_2/2)^2 = 169 - 25 = 144$ $d_2/2 = \sqrt{144}$ Since length must be positive, $d_2/2 = 12$ m. $d_2 = 12 \times 2 = 24$ m. Comparing the two diagonals, 10 m and 24 m, 24 m is the longer diagonal.
The perimeter of a rhombus is 52 m and its shorter diagonal is 10 m. The length of the longer diagonal is
Asked in BPSC Prelims Exam 67th Cancelled · Topic: General Concept (Spatial Distribution and Patterns (Fundamental Idea))
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