Question

The figure shows three right triangles. Triangles JKM, KLM, and JLK are similar.

Answer 1

Theorem: If two triangles are similar, the corresponding sides are in proportion.

Answer 2

Using the given theorem, which two statements help to prove that if segment JL is x, then x2 = 100?

A.Segment JL • segment JM = 64
Segment JL • segment LM = 48

B. Segment JL • segment JM = 48
Segment JL • segment LM = 36

C. Segment JL • segment JM = 64
Segment JL • segment LM = 36

D.Segment JL • segment JM = 36
Segment JL • segment LM = 64

Answer 3

anyone know how to do this?

Answer 4

do you know? @Calcmathlete

Answer 5

I know how to do the problem, but it's worded very strangely to me. It would make much more sense for x to be JK, KL, or KM. Give me a second, and I'll help you.

Answer 6

Ok. Have you done work with geometric means in these types of problems before?

Answer 7

no its actually a pretest so i havent done this before

Answer 8

but if x^2 is 100 then x must be 10

Answer 9

JL is 10 i mean

Answer 10

i thought it was x2 not x^2

Answer 11

Whenever you copy and paste, an exponent, it usually pastes it as x2 or something. @Venice-Gribbin

Answer 12

sorry hehe its x squared

Answer 13

i see.

Answer 14

Alright. Before we start, you may have noticed that you could've used Pythagorean Theorem to verify that JL = 10. Did you notice that?

Answer 15

yeah 36 plus 64=100

Answer 16

Ok. That's something that happens a lot in geometry. The key is that you need to use the given, which is the theorem they gave you. So, looking at similarity. The way the triangles are written are very significant. The order in which the letters appear tell you which sides are proportionate to what. So, if they were written as (JKM, KLM, and JLK) as you said, JL is proportionate to KL and JK. So, you could say that:\[\frac{JL}{KL}=\frac{JK}{KM}\]Are you following me so far?

Answer 17

yes so can we plug in their values into this proportion?

Answer 18

\[\frac{ 10 }{ 8 }=\frac{ 6 }{ ? }\]

Answer 19

If you have them, yes. Now, with this type of problem, you can only work with the values they gave you, even though it's obvious that JL does indeed equal 10. So, the only values you can actually plug in are JK and KL. The above proportion won't help you here. That was just for example. Hold on, and I'll help you get the right equation.

Answer 20

Lol @Gjallerhorn your name is perfect man

Answer 21

fellow destiny player?

Answer 22

Yeah I play it on xbox one. But before we continue this convo I want you to succeed in your math lol so ill wait till you get the answer

Answer 23

anyone who is familiar with Feizel crux's masterpiece knows their "Destiny..."

Answer 24

lolwut

Answer 25

I'll tell you how I got this specific proportion afterwards, but the one that will help you is\[\frac{JM}{JK}=\frac{JK}{JL}\]This can be simplified to \[JM\times JL=JK^2\]Now, plug in the values you have.

Answer 26

into the proportion or the equation under it?

Answer 27

The equation under it. I got the bottom equation by cross multiplying. The only values they gave you were JK and KL.

Answer 28

JMxJL=6^2 or 36?

Answer 29

Yes. That gives you the answer you need because none of the other answers have that as part of the answer. If you wanted to completely do the question, the other proportion you would use is:\[\frac{JL}{KL}=\frac{KL}{LM}\]which simplifies to \[JL\times LM=KL^2\]

Answer 30

ok lemme plug in

Answer 31

JLxLM=64 or 8^2

Answer 32

with those two values my answer would be D. correct?

Answer 33

Yes. That's correct. I'll write how I got those specific proportions in a second.

Answer 34

Now, the way I got those proportions is something called geometric means. A geometric mean between two numbers a and b is defined as \[Geometric ~Mean=\sqrt{ab}\]or you can also say \[ (Geometric~Mean)^2=a\times b\]Whenever you have a situation like this where you have the three similar right triangles with 2 of them within the bigger one, there are three relationships of "geometric means" that are extremely helpful (they are just similar triangles, so you can derive them later too).

Answer 35

One of the geometric mean relationships is that BD is the geometric mean of AD and DC. In other words, \[\frac{AD}{BD}=\frac{BD}{DC}\]or\[AD\times DC=BD^2\]which is a geometric mean. The other two are that BC is the geometric mean of CD and AC and that AB is the geometric mean of AD and AC.

Answer 36

I remember it with the pneumonic "ALL". That's the 'A' part of it.