```Question 542989
These three lines can form a triangle.
.
Here's a rule you can use. When you are given the lengths of three lines, add the lengths of the two short sides. If this sum is greater than the longest side, a triangle can be formed.
.
In this problem the two short sides are 6 and 10. Add them together and this sum is 16. Since 16 is greater than 15, a triangle can be formed.
.
Here's a way you can see why this rule works. Call the longest side Line AB. (In this problem scale the line AB so that the distance from A to B is 15 units.) Then set a drawing compass (the kind that you use to make a circle) so that the distance between the sharp point and the pencil lead is scaled to be the length of one of the short lines. (In this problem, you can set the compass so that it will make a circle that has a radius 6 units long.) Put the sharp point of the compass on one end of the longest line, say end A. Then draw a circle centered at point A with the radius equal to 6 units. Next set the compass so that it has a radius scaled for the other short side. (In this problem that would be 10 units.) Then put the point of the compass at end B of the longest line and draw the circle such that its center is point B and its radius is scaled to be the next to the shortest line. (Again, in this problem that circle will have a radius of 10 units.) If the two circles intersect in two places, you can make a triangle by drawing a line from A to an intersection point and then from B to that same intersection point. Notice that if the sum of the radii of the two circles are equal to or smaller than the length of the longest side, the two circles cannot intersect in two places because the circles are too small. Therefore, a triangle cannot be formed.
.
Think about this analysis or better yet, make up some examples and try them until you understand how this works.
.
And note that there are two special cases.
.
In the first case, if all the given lines are the same length (equilateral), a triangle can always be formed because if you add the lengths of any two of the lines, the sum will be longer than the third length.
.
In the second case, if two of the given lines are equal in length, add the lengths of these two equal sides. If this sum is greater than the third side, an isosceles triangle can be formed. But if this sum is equal to or smaller than the third side, a triangle cannot be formed. You can use the compass method to demonstrate this to yourself by setting the radius of the compass so that it is equal to the length of one of the two lines of equal length. Then call the third side Line AB and make two circles with the compass, on circle having A as its center, and the other having B as its center. If the two circles intersect at two places, a triangle can be formed. If the circles have a radius that is not long enough, the two circles will not intersect (they may even be tangent to each other) and a triangle cannot be formed.
.
Think about this and you may begin to see how this method of adding two of the sides and comparing them to the third side will allow you to determine if a triangle can be formed.
.
Hope this helps you to understand this problem.
.```