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Two sides of a triangle are AB=34cm and AC=25cm and their included angle measure 62°.
Find the distance of the orthocenter to side AB.
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In this problem, the given input data looks like is not related to the question,
so, it seems that it is unsolvable.
Perhaps, it is the reason why it remained unsolved about 5 years at this forum.
Nevertheless, the solution does exist and is quite beautiful, although not obvious, from the first glance.
So, we have a triangle ABC with the sides AB = 34 cm and AC = 25 cm.
Their included angle A is 62°, so we can find the length of the third side BC
opposite to angle A. Use the cosine law
BC = = = 31.3512096 cm.
Now, having the lengths of the three sides of triangle ABC, we can find its area,
using the Heron's formula. In order for do not bother with calculations, I will use
one of numerous online calculators,
https://www.omnicalculator.com/math/herons-formula
It gives the area of triangle ABC
area(ABC) = 375.253 cm^2.
Other online calculators
https://www.inchcalculator.com/herons-formula-calculator/
https://www.wolframalpha.com/widgets/view.jsp?id=7ac490665df1b278eb748160468147bc
give practically the same value.
Having the side lengths 'a', 'b' and 'c' of the triangle ABC, we can now
to determine the radius of the circumscribed circle around triangle ABC
R = = = 17.7537 cm (rounded).
Now the distance from the orthocenter to the side AB is the leg of the right angled triangle,
whose hypotenuse is R = 17.7537 cm and the other leg is half the length of the side AB.
So, we write
the distance from the orthocenter to the side AB = = 5.118 cm (rounded).
ANSWER. The distance from the orthocenter to the side AB is 5.118 cm (rounded).
Thus, all the data was woven into one logical thread that led to a complete solution.
So, we can celebrate the victory.