The heights AT and BD of triangle ABC intersect at point O. It is known that OT = 2cm

| April 28, 2021 | education

The heights AT and BD of triangle ABC intersect at point O. It is known that OT = 2cm, OD = 3cm, angle OAD = 30. Calculate the area of triangle ABC.

Consider a right-angled triangle ADO, in which, by condition, the angle of the AO is 30, then the leg OD lies opposite this angle and is equal to half the length of the AO hypotenuse. OD = AO / 2. AO = 2 * OD = 2 * 3 = 6 cm.

Consider two triangles, ADO and BTO. Both triangles are rectangular with right angles D and T. Angle AOD = BOT as vertical angles at the intersection of lines AT and BD.

Then:

OD / OT = AO / ВO.

3/2 = 6 / BO.

ВO = 2 * 6/3 = 4 cm.

Then the height BD = BО + DO = 4 + 3 = 7 cm.

Determine the area of ​​the triangle ABC.

From the right-angled triangle of the ATC, we determine the hypotenuse of the AC.

CosTAC = AT / AC.

AC = AT / Cos30 = (AO + OT) / Cos30 = 8 / (√3 / 2) = 16 / √3 cm.

S = AC * ВD / 2 = (16 / √3) * 7/2 = 56 / √3 cm2.

Answer: The area of ​​the triangle is 56 / √3 cm2.



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