The diagonals of the rhombus are 6 and 8. Find the ratio of the area of the inscribed circle to the area of the rhombus.
| August 18, 2021 | education
Knowing the lengths of the diagonals of the rhombus, we determine its area.
Savsd = АС * ВD / 2 = 8 * 6/2 = 24 cm2.
The diagonals of the rhombus divide it into four equal-sized right-angled triangles, then Saov = Saavsd / 4 = 24/4 = 6 cm.
The diagonals of the rhombus, at the point of their intersection, are halved and intersect at right angles.
Then AO = АС / 2 = 8/2 = 4 cm, ОВ = ВD / 2 = 6/2 = 3 cm.
AB ^ 2 = OA ^ 2 + BO ^ 2 = 16 + 9 = 25.
AB = 5 cm.
Saov = AB * OH / 2.
OH = 2 * Saov / AB = 2 * 6/5 = 2.4 cm.
OH = R = 2.4 cm.
Determine the area of the circle.
Sp = π * R2 = 5.76 * π cm2.
Then Samb / Savsd = 5.76 * π / 24 = 0.24 * π.
Answer: The area ratio is 0.24 * π
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