Triangles ABC and ABD are isosceles with base AB = 18 cm, base angles equal to 30 and 60

| May 7, 2021 | education

Triangles ABC and ABD are isosceles with base AB = 18 cm, base angles equal to 30 and 60 degrees, respectively. Find the angle between the planes of these triangles if the distance between C and D = √189.

In the data on the condition of isosceles triangles, we draw the heights (medians) of CO and DO.
Let’s use the definition of the tangent of an angle and find these heights.
In a right-angled triangle AOC:
Tg 30 ° = CO / AO → CO = AO * tan 30 ° = 9 * √3 / 3 = 3√3 (cm).
In a right-angled triangle AOD:
Tg 60 ° = DO / AO → DO = AO * tan 60 ° = 9 * √3 = 9√3 (cm).
Consider the triangle COD and write the cosine theorem in it:
CD² = CO² + DO² – 2 * CO * DO * cos COD = 27 + 243 – 162 * cos COD
189 = 27 + 243 – 162 * cos COD
162 * cos COD = 81
Cos COD = 1/2.
We conclude that the COD angle is 60 °.
Answer: the angle between the planes is 60 °.



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