The ratio of the lengths of the diagonals of the rhombus is 3: 4. Find the area of a rhombus if its side is 50 cm long.
| April 4, 2021 | education
1. Vertices of the rhombus A, B, C, D. О – the point of intersection of the diagonals AC and BD. BD: AC = 3: 4.
2. By the condition of the problem BD: AC = 3: 4. Therefore, BD = 3AC / 4.
3. The ВOС triangle is rectangular, since, according to the properties of the rhombus, its diagonals are mutually perpendicular.
5. BO² + CO² = BC² (by the Pythagorean theorem).
4. BO = BD / 2, CO = AC / 2.
BC = 50 cm (according to the problem statement).
5. ВD² / 4 + АC² / 4 = 2500. Substitute 3АС / 4 instead of ВD:
9АС² / 64 + АC² / 4 = 2500.
25АС² = 2500 x 64.
AC² = 6400.
AC = √6400 = 80 cm.
ВD = 3АС / 4 = 3 x 80/4 = 60 cm.
Answer: the diagonals AC and BD are 80 cm and 60 cm respectively.
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