Question 1151807: find a triangle two of whose angle have sizes TAN-1 (1.5) and TAN-1 (3). answer this question either by giving coordinates for three vertices, or by giving lengths of the three sides. To the nearest 0.1 degree, find the size of the third angle in your triangle.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the sum of the angles of a triangle is always equal to 180 degrees.
therefore, the third angle would be equal to 180 minus arctan(1.5) minus arctan(3) = 52.12501635 = 52.1 degrees rounded to the nearest tenth of a degree.
arctan(1.5) = 56.30993247 degrees.
arcan(3) = 71.56505118 degrees.
the third angle is equal to 180 minus those 2 angles = 52.12501635 degrees = 51.1 degrees rounded to the nearest tenth of a degree.
since you only know the angles, then the length of the sides can be any value as based on the height of the triangle.
assume that the triangle is ABC.
angle A is equal to arctan(1.5).
angle C is equal to arctan(3).
the height of the triangle is equal to BD.
since tan = opposite / adjacent side, then:
tan(A) = BD / AD = 1.5
solve for AD to get AD = BD / 1.5
tan(C) = BD / DC = 3.
solve for DC to get DC = BD / 3.
these ratios will always hold, regardless of the value of BD.
if you know the value of BD, then you can find the length of the side of the triangle.
for example:
if BD = 12, then:
AD = BD / 1.5 = 8.
DC = BD / 3 = 4.
AB = sqrt(8^2 + 12^2) = sqrt(208)
DC = sqrt(4^2 + 12^2) = sqrt(160)
AC = 8 + 4 = 12
if BD = 24, then all measurements will be doubled, since the second triangle formed is similar to the first triangle formed because all the angles of the second triangle are equal to the corresponding angles of the first triangle.
bottom line is you can have an infinite number of triangles whose sides are proportional and whose angles are equal.
my diagram shows you two such triangles.
the first is triangle ABC.
the second is triangle EFG.
the first triangle has a height of 12.
the second triangle has a height of 24.
both triangles have the same corresponding angles.
degrees of angles shown are rounded to 1 decimal place.
length of sides shown are rounded to 1 decimal places as well.
the intermediate calculations made, however, are not rounded, i.e. rounded to the number of decimal places that the calculator can hold.
triangle EFG is similar to triangle ABD.
that's because the corresponding angles are equal.
therefore, the corresponding sides are proportional.
you can see this in the diagram because the corresponding sides of triangle EFG are twice the length of the corresponding sides of triangle ABC.
here's my diagram.
not shown, but also relevant, is the fact that AC = 12 and EG = 24.
EG / AC = 24 / 12 = 2
those corresponding sides are proportional as well.
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