The perpendicular bisector \overline{AB} in the right triangle \triangle ABC

podnescijy 2021-11-25 Answered
The perpendicular bisector \(\displaystyle\overline{{{A}{B}}}\) in the right triangle \(\displaystyle\triangle{A}{B}{C}\) forms the triangle with the area 3 times smaller than the area of \(\displaystyle\triangle{A}{B}{C}\). Find the measures of acute angles in \(\displaystyle\triangle{A}{B}{C}\)

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Expert Answer

Supoilign1964
Answered 2021-11-26 Author has 5853 answers
Step 1
The Sides of AB and AC have unit length
Hence, \(\displaystyle\triangle{A}{B}{C}\) is isosceles with Congruent angles \(\displaystyle\angle{B}\) and \(\displaystyle\angle{C}\) NSNK Since \(\displaystyle{m}{\left(\angle{A}\right)}\) is given as 60 this means that,
\(\displaystyle{m}{\left(\angle{B}\right)}+{m}{\left(\angle{C}\right)}={120}\)
and so, \(\displaystyle{m}{\left(\angle{B}\right)}+{m}{\left(\angle{C}\right)}={60}\)
Therefore \(\displaystyle\triangle{A}{B}{C}\) is equilateral
Equilateral triangles have three lines of reflective symmetry in which the lines joining each vertex to the midpoint of the opposite side.
This means that CD is a line of symmetry for \(\displaystyle\triangle{A}{B}{C}\) and so CD is perpendicular to AB.
Step 2
By applying Pythagorean Theorem to the right triangle \(\displaystyle\triangle{A}{D}{C}\)
\(\displaystyle{\left|{A}{D}\right|}{2}+{\left|{C}{D}\right|}{2}={\left|{A}{C}\right|}{2}\)
We Know that D is the midpoint of AB.
B is on the unit circle,
Hence \(\displaystyle{\left|{A}{B}\right|}={1}\) and \(\displaystyle{\left|{A}{D}\right|}={12}\)
Since C is unit circle, We have \(\displaystyle{\left|{A}{C}\right|}={1}\)
Plugging the values for \(\displaystyle{\left|{A}{D}\right|}\) and \(\displaystyle{\left|{A}{C}\right|}\) into the formula gives,
\(\displaystyle{\left|{C}{D}\right|}={3}\sqrt{{{2}}}\). Since CD is perpendicular to AB .
Therefore, \(\displaystyle{C}{\left({12},\ {3}\sqrt{{{2}}}\right)}\)
\(\displaystyle{{\sin{{60}}}^{{\circ}}=}{3}\sqrt{{{2}}}\) and \(\displaystyle{{\cos{{60}}}^{{\circ}}=}{12}\)
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