Question

The last question I have I know I am dumb, but The thing is I'm better at Algebra :/

Answer 1

Idk why I can't set this up!

Answer 2

okay, can you tell me anything about the relationship between the small triangle and the big triangle?

Answer 3

The big and small triangles are similar.

Answer 4

They would be proportional to each other

Answer 5

Their base, and I guess you would call it a leg is congruent

Answer 6

uh, no, the bases are not congruent — the two segments that make up the base of the larger triangle are congruent...

Answer 7

Are you a girl?
Cause girls are better at algebra and guys are better at geometry.

Answer 8

I'm a counter-example to that theory :-)

Answer 9

Yes I am a girl but my BFF is way better at geometry but she's a lesson behind so she couldn't help!

Answer 10

9 times of 10 it's true though : )

Answer 11

I know the theory but I wasn't in class , and my teacher is really hard with her tests

Answer 12

and she didn't help me when I asked

Answer 13

those two segments labeled \(c\) are congruent, so we know that the ratio \(4:c\) is the same as the ratio of the hypotenuse of the big triangle to the base of the big triangle

can you write an equation for that?

Answer 14

4c = c(5x+2)
Try to solve it from there

Answer 15

Okay!

Answer 16

Sorry that would be
4(2c) = c(5x+2)

Answer 17

Okay so the first part is 8c - 1c = 8=5x+2 subtract two from both sides then you get 7 , so then you would divided by 5 and get 5/7 :/

Answer 18

no, that's still not the correct equation...

please, try to write the equation yourself first, you'll need to know how to do that to solve these in the future.

Answer 19

I thought it would be 5x-2 divided by 1/2= 4 , or somethign along that line

Answer 20

Here's the ratio of the hypotenuse to the base for the little triangle:

\[\frac{4}{c}\]

What is the ratio of the hypotenuse of the big triangle to the base of the big triangle?

Those two will be equal, so you can set them equal to each other, cross-multiply and solve for \(x\).

Answer 21

\(c\) will magically disappear without our ever having needed to know its value :-)

Answer 22

Would I cross mulitply like this X/5 *4/c ?

Answer 23

first you write the ratio I asked you for...

Answer 24

I apologize to my medicine is off I can't think clearly!

Answer 25

once you're a master at these problems you can start skipping steps, but first you have to become a master :-)

Answer 26

what is the hypotenuse of the big triangle?

what is the base of the big triangle?

Answer 27

You will need to cross multiply, but rethink what you are cross multiplying

Answer 28

Okay can you explain the ratio to me

Answer 29

5x/2, and 4/c ?

Answer 30

5x-2 on the big triangle is proportional to 4 on the little triangle
2c on the big triangle is proportional to c on the little triangle

Answer 31

Oh so it would be 5x-2/2c * c/4?

Answer 32

the hypotenuse of the big triangle is 5x-2
the base of the big triangle is the base of the little triangle plus another segment the same length, or c + c = 2c

that gives us:
\[\frac{4}{c} = \frac{5x-2}{2c}\]

Answer 33

Okay so now I cross multiply to get 5x-2*4
/2

Answer 34

would I multiply -2 to 4

Answer 35

Guys I'm sorry I;m so confused now.........

Answer 36

And you left :/

Answer 37

I got the hint ....... okay thanks anyways

Answer 38

\[\frac{4}{c} = \frac{5x-2}{2c}\]
Okay, if we cross-multiply, we multiply the numerator of one fraction by the denominator of the other, and vice-versa.

\[4*2c = c*(5x-2)\]\[8c = 5cx - 2c\]
Can you do the rest?

Answer 39

Yes add 2 c to the 8c which would be 10c=5cx and since they have a common variable then I can divided so x= 2?

Answer 40

\[8c = 5cx-2c\]\[8c+2c = 5cx-2c+2c\]\[10c = 5cx\]we can divide both sides by \(c\)
\[\frac{10c}{c} = \frac{5cx}{c}\]\[10 = 5x\]\[x=2\]

Now we do the important part, which is to check our answer!

If our hypotenuse on the big triangle is \(5x-2\) and we think \(x=2\), then the hypotenuse is \(5(2)-2 = 10-2=8\). Is that twice the length of the other hypotenuse?
Yes, it is, just like our base is twice the length of the other base. Our answer is correct.

Answer 41

Okay so it is definitely 2 thank you so much for putting up with me!

Answer 42

You're welcome!

By the way, I did the ratio as hypotenuse / base, but any way you want to do it is fine, so long as you do it the same way for both triangles.

Answer 43

Okay I'll keep that in mind thank you again!

Answer 44

@whpalmer4 you did a great job explaining it!

Answer 45

By the way my subjects I am good at is Biology, and English. The others nope!

Answer 46

@Hero you made some mistakes but got the right answer, and I think it is interesting to see why that is.

Small triangle you wrote:
\[a^2+b^2=4\](but it should be \(a^2+b^2=4^2 = 16\))

Large triangle you wrote:
\[4(a^2+b^2)=(5x-2)^2\]\[a^2+b^2=\frac{(5x-2)^2}{4}\]then
Notice: \(4 = \frac{(5x-2)^2}{4}\)
Solve for x
\[64 = (5x-2)^2\]
of course you can only get there from substituting the correct \(a^2+b^2=16\) earlier:

\[(2a)^2+(2b)^2 = (5x-2)^2\]\[4a^2+4b^2=(5x-2)^2\]\[4(a^2+b^2) = (5x-2)^2\]\[4(16) = (5x-2)^2\]\[64 = (5x-2)^2\]\[\sqrt{64} = \sqrt{(5x-2)^2}\]etc. Probably just wrote it down wrong after doing it correctly in your head. That's not the interesting part.

This approach using the Pythagorean theorem assumes that we have a right triangle, but the problem does not specify that it is! The similar triangles approach works for any pair of similar triangles where the legs are doubled in length, not just a right triangle. In fact, it is the natural of similar triangles that makes this work — double the length of the two congruent sides and you double the length of the third side, so the problem is always just \[2*4 = 5x-2\]\[x=2\]and even my approach could have been simplified.

You can write down the work as you did it for triangles which are not right triangles and even though the math is not correct (Pythagorean theorem will not apply), you will still get the same answer! All hail the power of similar triangles! Here is such a pair of triangles.