Vertices A and D of rhombus ABCD lie in the alpha plane. The distance from vertex B to this plane is 5 cm.

| May 7, 2021 | education

Vertices A and D of rhombus ABCD lie in the alpha plane. The distance from vertex B to this plane is 5 cm. Calculate the perimeter of the quadrangle whose vertices are points B, C and their projections onto the alpha plane (In centimeters) if the angle between side AB and its projection is 30 degrees.

BC will be parallel to AD.
AD will belong to the alpha plane, which means that BC will be parallel to the alpha plane.
If C1 is the projection of point C, CC1 lies in the alpha plane, and
B1 is the projection of point B, BB1 lies in the alpha plane, then CC1B1B is a rectangle, and C1B1 = CB = 8.
The distance you need to find is the leg of a right-angled triangle: x = BB1 = CC1.
According to the rule: the sum of the squares of the diagonals of the parallelogram will be equal to the doubled sum of the squares of its two adjacent sides.
DB² + AC² = 2 * (8² + 10²).
x² + DB1² = DB², therefore DB² = x² + 12².
x² + AC1² = AC² = AC² = x² + 6².
2 * (8² + 10²) = 2 * x² + 12² + 6².
2 * x² = 148.
x² = 74.
x = √74.



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