WHAT WILL BE JL

Question

What is JL?

JL =  cm

Answer 1

Hi Jessica -

Thanks for your question! To figure this one out, we need to use a theorem specific to this diagram, known as an "altitude-on-hypotenuse" diagram, in which drawing an altitude (KM) to the hypotenuse (JL) of a right triangle (triangle JKL) will divide that bigger right triangle into two smaller right triangles (JMK and KLM), such that the bigger right triangle and the two smaller right triangles are all similar to each other. If you have questions about how this theorem works, I'd be happy to dig deeper into why we know this is true during a session.

One consequence of this idea is the following useful theorem, which we will apply here:

The length of the altitude to the hypotenuse of a right triangle (KM) is the mean proportional (aka geometric mean) between the lengths of the two segments the altitude divides the hypotenuse into (JM and ML).

Mathetically, we express the idea that some number B is the mean proportional between two other numbers A and C using the following proportion (again, let me know if more questions):

A/B = B/C

Substituting in the names of the segments we have in this problem, we get:

JM/KM = KM/ML

Filling in the known lengths for the segments:

JM/6 = 6/4

After cross multiplying, we can solve the equation:

4 * JM = 36

JM = 9

Therefore the length of JM will be 9. Now that we know JM, we can add together the lengths of JM and ML to find the length of JL:

JM + ML = JL

9 + 4 = 13

This means that the length of JL will be 13. Let me know if any other questions, I hope this helps!