Question

The lengths of the sides in a right triangle form three consecutive terms of a geometric sequence. Find the common ratio of the sequence. There are two distinct answers.

Answer 1

A. 10.0 cm B. 34.6 cm C. 85.1 cm D. 120.0 cm which on would it be ?>

Answer 2

i need to find the common ratio not a length

Answer 3

Ok. so let the required ratio be R.

Then is it true that - the sides will be a, aR, aR^2 with aR^2 as the largest side i.e. hypotenuse?

Answer 4

i had a/r,a,ar but sure

Answer 5

@luckythebest

Answer 6

then is it true that - a^2 + a^2R^2 = a^2R^4 By pythagorean theorem?

Answer 7

yup

Answer 8

soooo

Answer 9

is it further true that -

a^2(1+R^2) = a^2R4

=> 1+R^2 = R^4 ?

Answer 10

Let the side lengths be \(1, a, a^2\) for some nonnegative number \(a\). If \(a > 1\), then \(a^2\) is the longest side so \((a^2)^2=(1)^2+(a)^2\). If \(0 < a < 1\), then \(1\) is the longest side so \((1)^2=(a)^2+(a^2)^2\). We can omit the \(a=1\) case since that would be an equilateral triangle. This should give you enough information to come up with the two distinct answers.

Answer 11

@yakeyglee who said 0<a<1

Answer 12

and yes lucky

Answer 13

Next - Let R^2 = K

Then is it true that -

1+K = K^2 ?

Answer 14

yup

Answer 15

We have -

K^2 - K - 1 = 0

Compare with ax^2 + bx + c = 0

a = 1, b = -1, c = -1

Find b^2-4ac and substitute it in the quadratic formula. Solve for K, and tell me the answer.

Answer 16

If we assume \(0<a<1\) that is one of the two cases. It affects how you treat the computation.

Answer 17

@yakeyglee if you substitute ^ in the quadratic formula you anyway get 2 distinct answers; example x+a and x-a, isn't it?

Answer 18

(1(+-)sqrt5)/2

Answer 19

Ok. so K = 1 +/- Sqrt 5/2

But we told earlier that K = R^2

Hence R^2 = 1 +/- sqrt 5/2

R = Sqrt of R^2.

The answer is a bit complex but that is how its done. :(

Answer 20

so r=sqrt((1+/- sqrt5)/2)

Answer 21

:like:

Answer 22

?

Answer 23

i mean its correcto :)

Answer 24

k thanks

Answer 25

If you want further simplification sqrt 5 = 2.236

then R = (1+2.236/1.414)

=> sqrt of 3.236 / 1.414 (1.414 is sqrt of 2)

Answer 26

@luckythebest I didn't use any x in my computation... i don't understand what you are talking about. And you will get multiple values for \(a\), but I bet only one of them satisfies the initial assumption that was made to derive the equation.

Answer 27

@yakeyglee what i meant to say was that if you use the quadratic equation formula anyway you will get two distinct answers.