In the rectangle ABCD, CF and AE are the perpendiculars to the diagonal BD. Angle between diagonals = 30 °

| June 2, 2021 | education

In the rectangle ABCD, CF and AE are the perpendiculars to the diagonal BD. Angle between diagonals = 30 °, CF = 2cm. Prove that AE = CF. Find the diagonal BD (length).

Since the segments AE and CF are perpendicular to BD, the triangles AOE and COF are rectangular.

The diagonals of the rectangle, at the point of intersection, are divided in half, then AO = CO. Angle COF = AOE as vertical angles.

Then the right-angled triangles AOE and COF are equal in hypotenuse and acute angle, and therefore AE = CF, which was required to prove.

In a right-angled triangle COF, the CF leg lies opposite an angle of 30, then OC = 2 * CF = 2 * 2 = 4 cm.

Then AC = 2 * OC = 4 * 2 = 8 cm.

The diagonals of the rectangle are equal, then BD = AC = 8 cm.

Answer: The length of the diagonal BD is 8 cm.



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