Question 1120894: The figure shows a triangle STU where TU=11cm. H lies on TU such that the length of TH is 120% of the length of HU and angle SHU=90degree. Given that the area of triangle STH is 21cm^2, find angle TSU.
Found 2 solutions by solver91311, Theo:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Since the measure of the base is 11 and the area is 21, the height, which is to say segment SH must be  .
Further, since TH is 120% of HU, let the value HU be designated as  , then we can say:
For the sake of clarity, designate angle TSH as  and angle USH as  .
By definition, 
Likewise, 
The tangent of the sum of two angles is given as \ =\ \frac{\tan{a}\ +\ \tan{b}}{1\ -\ \tan{a}\tan{b}}) , so:
\ =\ \frac{\frac{66}{42}\,+\,\frac{55}{42}}{1-\(\frac{66}{42}\)\(\frac{55}{42}\)}\ \approx\ -2.7235)
Since Angle TSU is clearly the sum of angle plus angle :
\)\ \approx\ \tan^{-1}(-2.7235))
Your calculator will give you a negative angle as an answer. Providing you have the calculator set to work in degrees, add 180 degrees to the calculator result to get the appropriate QII positive number result.
John
My calculator said it, I believe it, that settles it
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! your triangle looks like this:
TU = 11 cm.
TH = 120% of HU.
if we let x = HU, then 1.2x = TH
TH + TU = 11 results in 1.2x + x = 11 which results in 2.2x = 11.
solve for x to get x = 11/2.2 = 5
that makes 1.2x = 1.2 * 5 = 6
since 1.2x = TH, we have TH = 6 and we have HU = 5.
the area of triangle STH is equal to 21 cm^2.
area of a triangle is equal to 1/2 * base * height.
the area is 21 and the base is 6.
formula becomes 21 = 1/2 * 6 * H, where H is equal to SH.
solve for H to get H = 21 * 2 / 6 = 7.
the height of the triangle is equal to SH which is equal to 7.
we are looking for the measure of angle TSU.
the measure of angle TSH is equal to arctan(TH / SH which is equal to arctan(6/7).
that makes angle TSH equal to 40.60129465 degrees.
the measure of angle HSU is equal to arctan(HU/SH) which is equal to arctan(5/7).
that makes angle HSU equal to 35.53767779.
angle TSU is the sum of these two angle.
that makes angle TSU equal to 40.60129465 + 35.5376777 = 76.13897244 degrees.
that's your solution.
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