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In triangle $ABC$, $\angle A = 30^\circ$ and $\angle B = 60^\circ$. Point $X$ is on side $\overline{AC}$ such that line segment $\overline{BX}$ bisects $\angle ABC$. If $AB = 12$, then find the area of triangle $BXA$.

In triangle $ABC,$ point $D$ is on $\overline{AC}$ such that $AD = 3CD = 12$. If $\angle ABC = \angle BDA = 90^\circ$, then what is $BD$?

Find the area of triangle $PQR$ if $PQ = QR = 12$ and $\angle PQR = 120^\circ$.

In triangle $ABC,$ $AB = 10,$ $BC = 24,$ and $AC = 26.$ Find the length of the shortest altitude in this triangle.

Jul 13, 2020

#1
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First problem: By the angle bisector theorem, the area of triangle BXA is 15*sqrt(2).

Second problem: BD^2 = AD*CD = 3*12 = 36, so BD = 6.

Jul 13, 2020
#2
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Both of these are incorrect

qwertyzz  Jul 13, 2020
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This will help you with the latex:

https://artofproblemsolving.com/wiki/index.php/LaTeX

Jul 13, 2020