Two sides of a triangle have lengths 7 and 13. The third side has a length that is ______.

Question

 This is geometry and triangles

Answer 1

Greetings Camryn from Madison,

Thank you for your question.

You asked if we are given the length of two sides of triangle, 7 and 13, what can we say about the third side.

There is a rule for every triangle known as the triangle inequality which states:

Triangle Inequality: The length of any side of any proper triangle is less than, the sum of the other two.

For a triangle with sides whose lengths are given as positive numbers a, b, c; this gives 3 inequalities:

a+b>c, b+c>a, c+a>b.

which can also be written as

|a-b|<c<a+b.

Note: if you include degenerate triangles where all vertexes are on the same line (collinear,) then these inequalities can be equalities, but as degenerate triangles are not technically triangles, I didn't include them.

Back to your question, given a = 7 and b = 13, this gives us for c:

6 < c < 20,

Of course if you knew the triangle was a right triangle you could use the Pythagorean theorem to directly get the size of the third side.

Hope this helps.

Answer 2

The length must be less than 20, and greater than 6.

 

6 < L < 20