Question

How many triangles of integral sides are possible such that two of its sides are of length 10 units and 7 units ?

Answer 1

consider the smallest possible side ,
and the largest

Answer 2

Recall: The length of the triangle sides must satisfy the triangle inequality.
Let say, the missing side is x.
By triangles inequality, we have sum of two sides must the greater than the third. So, we have three inequalities. They are
10 + 7 > x and
7 + x > 10 and
10 + x > 7
Now, it would be better for you to draw a number line and mark down the solutions of these inequalities. The integral overlapping part would be your solution.

Answer 3

i dont understand that third inequation @MonoPolyRoly

Answer 4

The triangle inequality states that the sum of the lengths of ANY two sides of a triangle is greater than the length of the remaining side.[1] So, we need to test all the possible cases, i.e. 3C2 = 3 cases.

Suppose we can form a triangle with side of length being 7, 10 and x, where x is a non-negative number. We can draw the triangle as follows:


By triangle inequality, we can set up the THREE inequalities. x must satisfy the three inequalities such that it can form a triangle with sides of length being 7 and 10. The inequalities are
10 + 7 > x and
7 + x > 10 and
10 + x > 7

We just set up the inequalities, but we have to check if there is such x satisfying all the inequalities above, and of course, x must be non-negative. If we can find such x to exist, we may try to look for the integral solution; else, we can conclude that no triangle can be formed with given length being 7 and 10.

Answer 5

The triangle inequality states that the sum of the lengths of ANY two sides of a triangle is greater than the length of the remaining side.[1] So, we need to test all the possible cases, i.e. 3C2 = 3 cases.

Suppose we can form a triangle with side of length being 7, 10 and x, where x is a non-negative number. We can draw the triangle as follows:


By triangle inequality, we can set up the THREE inequalities. x must satisfy the three inequalities such that it can form a triangle with sides of length being 7 and 10. The inequalities are
10 + 7 > x and
7 + x > 10 and
10 + x > 7

We just set up the inequalities, but we have to check if there is such x satisfying all the inequalities above, and of course, x must be non-negative. If we can find such x to exist, we may try to look for the integral solution; else, we can conclude that no triangle can be formed with given length being 7 and 10.

Reference:
1. Triangle Inequality, WolframAlpha.
Retrieved from http://mathworld.wolfram.com/TriangleInequality.html