The diagonals of the rectangle ABCD intersect at point O, the angle CDO is 60 degrees, AC = 10cm.

| September 19, 2021 | education

The diagonals of the rectangle ABCD intersect at point O, the angle CDO is 60 degrees, AC = 10cm. Find the perimeter of the triangle OCD.

In a rectangle, the diagonals, by their intersection, form two pairs of isosceles triangles. Consider a triangle CDO, in which, by condition, the angle at the base is known, the angle CDO = 60 °, respectively, and the second angle at the base, the angle DCO = 60 °. The COD angle at the apex of this triangle is 180 ° – (60 ° + 60 °) = 60 °. The CDO triangle is equilateral.
Consider the triangle AOD, in it:
Angle OAD = Angle ODA = 90 ° – Angle CDO = 90 ° – 60 ° = 30 °

Consider a right-angled triangle ADC, in which we know by condition the hypotenuse AC = 10 cm and the angle OAD = 30 °. The CD leg is located opposite an angle of 30 °, which means that it is equal to half of the AC hypotenuse, CD = 1/2 AC = 1/2 * 10 = 5 cm.
Find the perimeter of an equilateral triangle OCD:
P = 3 * a = 3 * 5 = 15 cm.
Answer: 15 centimeters.



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