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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

Minezap90
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  • Welcome to MSE! Can you [edit] your question to include your thoughts and efforts on this problem? What have you tried, and where are you having difficulty? This will help people write an appropriate answer the addresses your problem. Questions that include this information tend to have a much better response. –  Jan 21 '16 at 00:45
  • You say "L + 10 = Largest Side" Then what is L? What plus 10 = Largest Side? Is L a good label for the number. Likewise you say "M*3 = the 3rd Side". What is M? What times 3 = the 3rd side? Is M the right label. (Are L and M the same thing?; after all, you add 10 to the shortest side and you multiply the shortest side by 3..) – fleablood Jan 21 '16 at 01:06
  • having discovered the system to solve, I can't help but think the question itself is flawed...... consider K. Jiang's answer... if $a$ was the longest side, why would it be smaller than $b$? Where does this question arise? – Eleven-Eleven Jan 21 '16 at 01:17

3 Answers3

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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 $\text{NO SOLUTION}.$

K. Jiang
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  • interesting that a, the "longest side" is less than half the size of $b$... – Eleven-Eleven Jan 21 '16 at 01:14
  • It is irrelevant that $a < c$; the equations are correct and all we care about is the side lengths, not what we choose to name them. @Eleven-Eleven – K. Jiang Jan 21 '16 at 01:18
  • Thx for the answer I needed the help I was about to drive myself crazy – Minezap90 Jan 21 '16 at 01:18
  • And that's what i believe too, K. Jiang... I don't dispute the answer, just the question and why the problem returns the side that is supposed to be in between the largest and smallest as the largest. That is a problem... – Eleven-Eleven Jan 21 '16 at 01:23
  • Also, the solution triangle does not close. You can't get back from the ends of the 69 side with only 56 (23 + 33) total length... – DJohnM Jan 21 '16 at 01:25
  • I see what you mean now... OP claimed that the smallest side plus $10$ equals greatest side, and the greatest side is actually the "other side"... oh well, we are never safe from bad problems. Thanks for clarifying @Eleven-Eleven – K. Jiang Jan 21 '16 at 01:25
  • Thank you; I have edited me solution. @DJohnM – K. Jiang Jan 21 '16 at 01:29
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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

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

Eleven-Eleven
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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.

fleablood
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