Triangle Perimeter 42 cm. Long side 2 cm. less than sum of the other sides. Twice the shortest side is 10 cm. less than the longest side. Find length of sides.

Question

I would derive 3 equations from this problem. I already know the answers from the shortest angle to the longest are:

5 cm., 17 cm., 20 cm. but I haven't been able to get these answers.

Answer 1

a=short side,  b= side 2 and c=long side

a+b+c=42 cm , Equation (1)

c=a+b-2  Equation (2)

2a=c-10  Equation (3)

 

Solve by 1st solving for a+b=c+2 from (2) and substituting into (1) giving c+2+c=42  or c=40/2=20 cm

Next;  From (3) substitute for c and solve for a : 2a=20-10,  a=10/2=5 cm

Finally solve for b from (1) or (2). Using (1)   5+b+20=42, b=42-20-5=17cm

Answer 2

If you have 3 variables, you need 3 equations to solve.

 

I started by drawing a triangle, labeling the sides x (shortest), y (longest), and z (middle).  

 

Equation 1: Perimeter is the sum of all sides.
x+y+z=42

Equation 2: The trick is getting the 2 correctly.
y=x+z-2.  I rewrote this equation to be: x+z=y+2

 

Equation 3: 2x=y-10

 

Now, I substitute equation 2 into equation 1.  This will give me 1 equation with 1 variable.

x+y+z=42
y+(y+2)=42
2y+2=42
2y=40

y=20

 

Now that you know y, it's simple to substitute into the other equations and solve for x and z.