How is $\cot(J)$ defined in terms of the adjacent and opposite sides of angle $J$? Write a fraction using $JK$ and $KL$ that equals $\dfrac{7}{24}$.

Once you know that $JK$ and $KL$ are in the ratio $7:24$, try writing $JK = 7x$ and $KL = 24x$, then use the Pythagorean theorem with hypotenuse 20 to find $x$.

Substitute your expressions for $JK$ and $KL$ into $JL^2 = JK^2 + KL^2$, use $JL = 20$, and solve for $x$. Then plug back in to find $JK$.

In Desmos, enter the expression 7*(20/25). This multiplies the leg 7 from the $7$–$24$–$25$ triangle by the scale factor that changes the hypotenuse from 25 to 20. The numerical output is the length of $JK$.

Right triangle $JKL$ has a right angle at $K$, so $JL$ is the hypotenuse.

At angle $J$:

• Adjacent side: $JK$
• Opposite side: $KL$
• Hypotenuse: $JL = 20$

By definition, $\cot(\theta) = \dfrac{\text{adjacent}}{\text{opposite}}$.

So

This means the legs are in the ratio

• $JK : KL = 7 : 24$.

Let

• $JK = 7x$
• $KL = 24x$ for some positive number $x$.

By the Pythagorean theorem:

Substitute $JK = 7x$ and $KL = 24x$:

So

We are told $JL = 20$, and from the previous step $JL = 25x$.

So

Solve for $x$:

Recall $JK = 7x$. Substitute $x = \dfrac{4}{5}$:

Therefore, the length of $JK$ is $\dfrac{28}{5}$.