geometry perimeter for triangles

Question

I don't get.. The largest side of the triangle (side a) is 10 more units that the smallest side (side b) and the 3rd side of the triangle(side c) is triple the smallest side of the triangle. if the perimeter is 125 units how many units is each side.

A+10=The perimeter of the Largest side of the triangle B*3=the Perimeter of the 3rd side Of the triangle C=the Perimeter of the smallest side of the triangle P= 125 units

Answer 1

Use algebra to solve this. Let $a$ be the longest side, $b$ the middle-length side, and $c$ the shortest side. We can write the following three equations: $$a = c + 10,$$ $$b = 3c,$$ $$a + b + c = 125.$$

To solve this system, substitute the first two equations in terms of $c$ into the third equation. This yields $5c + 10 = 125,$ so $c = 23.$ It follows that $a = 33$ and $b = 69.$

However, notice that $a < c,$ which leads to a contradiction. In addition, the three lengths found here cannot form a triangle, as $a + c = 56 < b = 69.$ Therefore, there is $\text{NO SOLUTION}.$

Answer 2

If $a$=the shortest side, $c$ the longest, and $b$ the middle, we have $$a+10=c$$ $$b=3c$$ $$a+b+c=125$$ You can use a matrix to find solutions or just use substitution to solve for a variable and back substitute for the remainig sides.

Note all equations have $c$, so solve for $c$ first since

$$a+b+c=10\Rightarrow c-10+3c+c=125\Rightarrow 5c-10=125...$$

Answer 3

Appearently there is no solution:

The longest side is ten more than the short side: $S + 10 = L$

The middle side is 3 times the shortest side: $M = 3S

The Perimeter is 125: $S + M + L = 125$

$S + (3S) + (S + 10) = 125$

$5S + 10 = 125$

$5S = 115$

$S = 23$; shortest side.

$M = 3*S = 3*23 = 69$; middle side.

$L = S + 10 = 23 + 10 = 33: long side.

But middle side > largest side is impossible.

So... Maybe the "the third side" isn't the middle side but actually the longest side:

$S + 10 = L$

$L = 3*S$

$S + L + M = 125$

So

$L = S + 10 = 3*S \implies 2S = 10 \implies S = 5$

$L = S + 10 = 15$

$S + L + M = 125 => 5 + 15 + M = 125 \implies M = 100$

So again middles side > largest side. (Not to mention; if one side is 10 and the other 5, the third side must be less than 15.)

So... the question is impossible to answer.