Find the Perimeter of a Right Triangle Given a Tangent Ratio

Expert intro: When one acute angle has a tangent value, you can turn that ratio directly into the two legs of the right triangle. After that, the hypotenuse is just the Pythagorean theorem.

Detailed walkthrough

In right triangle $XYZ$, angle $Z$ is $90^\circ$, so $XY$ is the hypotenuse.

Given:

• $YZ=24$
• $\tan X=\frac{12}{35}$

For angle $X$,

$\tan X=\frac{\text{opposite}}{\text{adjacent}}=\frac{YZ}{XZ}$

So

$\frac{24}{XZ}=\frac{12}{35}$

Cross-multiply:

$12\cdot XZ=24\cdot 35$

$XZ=\frac{24\cdot 35}{12}=70$

Now find the hypotenuse using the Pythagorean theorem:

$XY=\sqrt{24^2+70^2}=\sqrt{576+4900}=\sqrt{5476}=74$

Perimeter:

$24+70+74=168$

Answer: 168

💡 Pitfall guide

Students often mix up which leg is opposite or adjacent. Here, relative to angle $X$, side $YZ$ is opposite and side $XZ$ is adjacent. Also, do not skip the Pythagorean step after finding the missing leg; the perimeter needs all three sides.

🔄 Real-world variant

If the tangent ratio were simplified differently, the process would be the same: use the ratio to solve for the unknown leg, then compute the hypotenuse. If the right angle were at a different vertex, you would just relabel the opposite and adjacent sides before setting up the tangent equation.

🔍 Related terms

tangent ratio, Pythagorean theorem, right triangle perimeter