Why is the height of the smaller triangle 3cm?

senseimichael

senseimichael

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I have the answer key, and it says that the height of the smaller triangle is 3cm, but for the life of me, I could not see how we derive the 3cm for its height! Anyone can help?
 
Hello, and welcome to FMH! :)

Can you see the smaller triangle is similar to the larger one, and that we may state:

[MATH]\frac{4}{24}=\frac{h}{10}[/MATH]
But, this does not lead to \(h=3\).
 
I understand that there is a lesser known rule, whereby if all the sides are integers, the Pythagoras Theorem would give me a 3-4-5 ratio of sides. I just got this explained by a friend's middle school nephew! Some questions are like...set out to kill students! I do not remember having to do such questions when I was in school!
 
senseimichael said:
I understand that there is a lesser known rule, whereby if all the sides are integers, the Pythagoras Theorem would give me a 3-4-5 ratio of sides. I just got this explained by a friend's middle school nephew! Some questions are like...set out to kill students! I do not remember having to do such questions when I was in school!
Click to expand...
"...if all the sides are integers, the Pythagoras Theorem would give me a 3-4-5 ratio of sides....."

There is NO such rule.
 
senseimichael said:

It's called a Pythagoras triple.
Click to expand...
Yes there are many Pythagorean triples. Some of those are:

12, 5, 13

7, 24, 25

........
Actually there are "infinite" number of these triplets. They all refer to the sides of RIGHT-ANGLED triangles. You did not mention right-angled triangle in your statement. Moreover, as you can see 3-4-5 is the most "known" ratio - not the only ratio.
 
The answer key is utterly wrong; maybe the nephew is trying to justify the wrong answer.

Do you notice that the large triangle has sides 10, 24, 26? This is another Pythagorean triple, in the ratio 5:12:13. The 3:4:5 triangle has a different angle.

But the fact is that the legs of a right triangle do not have to be whole numbers at all. As has been shown, the height is actually 5/3 (the solution of the proportion in post #2).
 
Subhotosh Khan said:
Yes there are many Pythagorean triples. Some of those are:

12, 5, 13

7, 24, 25

........
Actually there are "infinite" number of these triplets. They all refer to the sides of RIGHT-ANGLED triangles. You did not mention right-angled triangle in your statement. Moreover, as you can see 3-4-5 is the most "known" ratio - not the only ratio.
Click to expand...

You are right. I was just explaining one part of his answer to me. I should have written out everything.
 
Dr.Peterson said:
The answer key is utterly wrong; maybe the nephew is trying to justify the wrong answer.

Do you notice that the large triangle has sides 10, 24, 26? This is another Pythagorean triple, in the ratio 5:12:13. The 3:4:5 triangle has a different angle.

But the fact is that the legs of a right triangle do not have to be whole numbers at all. As has been shown, the height is actually 5/3 (the solution of the proportion in post #2).
Click to expand...

The answer key was from the school.

You cannot assume the proportion, because you cannot assume the angle formed is the same as the bigger triangle's angle.
 
senseimichael said:
The answer key was from the school.

You cannot assume the proportion, because you cannot assume the angle formed is the same as the bigger triangle's angle.
Click to expand...
Of course you can! They are the very same angle! That is, the rays containing the sides 24 and 26 in the large triangle are the same rays containing the hypotenuse and long leg of the small triangle. The two triangles are similar, because any two right triangles with the same acute angle are similar.

On the other hand, there is no reason to assume that any given right triangle is a 3-4-5 triangle, or that its sides are integers, as you assume in post #3.

Teachers can be wrong, especially in answer keys.
 
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