Applying Pythagoras backwards (with minimal information)

[NoviceBM]

Hello all!

While working on a personal project, I encountered the following math problem. I have a triangle with sides A, B, and C. I know the length of C = 15,3. Furthermore, I know that B / A = 7, so I can conclude that B = 7A. The triangle is a right angled triangle. To illustrate the problem, I have uploaded a visual representation attached to this post.

My goal is to know the lengths of sides A and B.

Is it possible to calculate those, and if so, how? It seems like a simple problem to solve with Pythagoras, but it's been years since I last touched any formulas. I've been breaking my head over it for the past few hours and just can't seem to get a (correct) answer. Do I need more information to calculate this? Any help would be greatly appreciated!

 

[MathLover1]

so, c^2=a^2+b^2

if given:
c = 15,3
b / a = 7=> b = 7a

we have

15.3^2=a^2+(7a)^2........solve for a
15.3^2=a^2+49a^2
15.3^2=50a^2
15.3^2/50=a^2
a=sqrt(15.3^2/50)
a=15.3/sqrt(50)
a=2.1637467504308354

b = 7a =>b = 7*2.1637467504308354=15.1462272530158478

exact solution:
a=2.1637467504308354
15.1462272530158478

we can round it to one decimal place, so sides a and b are:
a=2.2 and b=15.2

 

[nycmathguy]

so, c^2=a^2+b^2

if given:
c = 15,3
b / a = 7=> b = 7a

we have

15.3^2=a^2+(7a)^2........solve for a
15.3^2=a^2+49a^2
15.3^2=50a^2
15.3^2/50=a^2
a=sqrt(15.3^2/50)
a=15.3/sqrt(50)
a=2.1637467504308354

b = 7a =>b = 7*2.1637467504308354=15.1462272530158478

exact solution:
a=2.1637467504308354
15.1462272530158478

we can round it to one decimal place, so sides a and b are:
a=2.2 and b=15.2

Very nice.

 

[NoviceBM]

so, c^2=a^2+b^2

if given:
c = 15,3
b / a = 7=> b = 7a

we have

15.3^2=a^2+(7a)^2........solve for a
15.3^2=a^2+49a^2
15.3^2=50a^2
15.3^2/50=a^2
a=sqrt(15.3^2/50)
a=15.3/sqrt(50)
a=2.1637467504308354

b = 7a =>b = 7*2.1637467504308354=15.1462272530158478

exact solution:
a=2.1637467504308354
15.1462272530158478

we can round it to one decimal place, so sides a and b are:
a=2.2 and b=15.2

Thank you so much! This is a very clear explanation.

I tried replicating the process with some different C and B/A values to see if I understood what was happening, and now I consistently managed to solve the equation. Turns out I did something wrong squaring my combined A and B values previously, which messed up the formula. Much appreciated!

 

[MathLover1]

when doing that kind of problems, make sure to recognize what is hypothenuse, what are legs, and which one is to find
formula you need is: c^2=a^2+b^2 where c is hypothenuse, a and b are legs

it means

the biggest square has the exact same area as the other two squares put together

an example:

Let's check if the areas are the same:

3^2 + 4^2 = 5^2

Calculating this becomes:

9 + 16 = 25

25=25

 

[nycmathguy]

when doing that kind of problems, make sure to recognize what is hypothenuse, what are legs, and which one is to find
formula you need is: c^2=a^2+b^2 where c is hypothenuse, a and b are legs

it means
View attachment 3353

the biggest square has the exact same area as the other two squares put together

View attachment 3354



an example:

View attachment 3355

Let's check if the areas are the same:

3^2 + 4^2 = 5^2

Calculating this becomes:

9 + 16 = 25

25=25

This is a math professional reply. A job well-done!