Arithmetic Series (1 Viewer)

[SunnyScience]

Fitzpatrick ex 11(b) Q40

The lengths of the sides of a right angled triangle form the terms of an arithmetic sequence. Of the hypotenuse is 15 cm in length, what is the length of the other two sides?

I used Pythagoras theorem and the quadratic formula, but the answer I got was 15 :/ (went in a circle)

Help please? c:

 

[SunnyScience]

Bump, please?

 

[nightweaver066]

 

[Shadowdude]

The side lengths are: a, a+d, a+2d. This is because they're in a geometric arithmetic sequence.

We are given a+2d = 15, because the hypotenuse is the longest side.

We use Pythagoras:







Now we know that a+2d = 15.

We note that:



And so, we use that fact to get:







Now we note that a = 15 - 2d. So:











So d = 15, or d = 3.

Try d=15 into what we know: a = 15 - 2d.



We cannot have a negative side value, so d cannot be 15. Thus, d = 3.

Try d = 3.



Our side lengths are a, a+d, a+2d. Or, 9, 12 and 15.

Check:



So we're right. The answer is 9 and 12.

 

[Drongoski]

The Pythagorean Triples: {3,4,5}, {5,12,13},{8,15,17} . . .

Obviously this one comes from the {3,4,5} family. For this question: the triples are 3 x {3,4,5} = {9,12,15}

I know the question expects you to work it out the usual way.

 

[SunnyScience]

Thanks guys for your help did some incorrect substitution ^^

Shadow: *arithmetic sequence

Drong: thanks, I knew that, but thank you for sharing as someone else doing this question might have not. ty

 

[Shadowdude]

Arithmetic sequence, yeah, you're right.

Point is, we got the right answer... eventually.

 

[SunnyScience]

Thanks for your time - I really do appreciate it

 

[Carrotsticks]

The Pythagorean Triples: {3,4,5}, {5,12,13},{8,15,17} . . .

Obviously this one comes from the {3,4,5} family. For this question: the triples are 3 x {3,4,5} = {9,12,15}

I know the question expects you to work it out the usual way.

This is assuming the solutions are integers.

Is it possible to have non-integer solutions satisfying an AP and Pythagoras' Theorem?

 

[Rezen]

Yes,
In-fact all scaling of the triple {3,4,5} by a real number form an AP. Since r x {3,4,5} = {3r,4r,5r}={3r, 3r + r, 3r + 2r}.
i.e, when r = 1.5, then the triple {4.5,6,7.5} satisfy both the Pythagorean theorem and form an AP.