MHB Finding the lengths and altitudes of s triangle

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For triangle ABC, given sin A = 12/13 and tan B = 15/8, the altitude to side AB is 60 units. The length of side AC is calculated to be 65 units, while side BC measures 68 units. The length of side AB is determined to be 57 units. The area of the triangle is found to be 1710 square units, leading to altitudes from points A and B to sides BC and AC measuring approximately 52.6 units and 50.3 units, respectively. The calculations demonstrate the relationships between the sides and altitudes effectively.
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For a certain triangle ABC, sin A = 12/13, tan B = 15/8, and the altitude to side AB is 60 units. Find the lengths of the sides and of the altitudes of the triangle.

Again, I couldn't come up with an image of the triangle in question. Maybe you can provide me the picture of the problem so that I can solve it. Thanks!
 
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Here is what I would begin with:

View attachment 3418

Now, if $$\sin(A)=\frac{12}{13}$$, then what must side $\overline{AC}$ be?
 

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Using $\csc(A)=\frac{AC}{60}$

$\frac{13}{12}=\frac{AC}{60}$

AC = 65 units
 
paulmdrdo said:
Using $\csc(A)=\frac{AC}{60}$

$\frac{13}{12}=\frac{AC}{60}$

AC = 65 units

Correct! :D

Now, can you use $$\tan(B)=\frac{15}{8}$$, along with what you already have to get $\overline{AB}$ and $\overline{BC}$, then use the area of the triangle to get the other two altitudes?
 
yes I can do that.

Solving for $\overline{BC}$

$\csc(B)=\frac{\overline{BC}}{60}$

$\frac{17}{15}=\frac{\overline{BC}}{60}$

$\overline{BC}=68$units

for the length of line segment from point B to the intersection of the perpendicular line and $\overline{AB}$

$\sqrt{68^2-60^2}=32$

And for the length of line segment from point A to the intersection of the perpendicular line and $\overline{AB}$

$\sqrt{65^2-60^2}=25$

hence,

$\overline{AB}= 25+32=57$unitsI have no Idea where those two altitudes are.
 
Last edited:
paulmdrdo said:
yes I can do that.

Solving for $\overline{BC}$

$\csc(B)=\frac{\overline{BC}}{60}$

$\frac{17}{15}=\frac{\overline{BC}}{60}$

$\overline{BC}=68$units

for the length of line segment from point B to the intersection of the perpendicular line and $\overline{AB}$

$\sqrt{68^2-60^2}=32$

And for the length of line segment from point A to the intersection of the perpendicular line and $\overline{AB}$

$\sqrt{65^2-60^2}=25$

hence,

$\overline{AB}= 25+32=57$units

Yes, I agree with those side lengths. :D

Now, compute the area of the triangle, using $\overline{AB}$, and its given altitude, then use this area to find the altitudes relative to the other two sides. :D
 
Hmm...

this is what I tried

$A=\frac{1}{2}{57}{60}=1710$ sq. units -->

from the result above solving for the altitude from point A to $\overline{BC}$

$1710=0.5(h)(68)$

$h=$50.29 units

solving for the altitude from point B to $\overline{AC}$

$1710=0.5(h)(65)$

$h=$ 52.6 units

Thank you very much! I couldn't have done this, if not for your help.
 
Yes, we can write:

$$h=\frac{2A}{b}=\frac{3420}{b}$$

So, relative to $\overline{AC}$, we have:

$$h=\frac{3420}{65}=\frac{684}{13}\approx52.6$$

And relative to $\overline{BC}$, we have:

$$h=\frac{3420}{68}=\frac{855}{17}\approx50.3$$
 
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