Suppose \(\triangle TIP\) and \(\triangle TOP\) are isosceles triangles. Also suppose that TI=5, PI=7 and PO=11.
What is the sum of all possible lengths for TP?
What is the sum of all possible lengths for TO?
My best illustration:
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TI = 5, PI = 7, PO = 11.
Since they are both isosceles triangles (2 sides are equal), and TI is not equal to PI, then TP has to be equal to TI or PI, meaning TP = 5, 7. Their sum is 12.
Now we know that TP = 5 or 7, which means TP is not equal to PO. Because of this, TO has to be equal to either TP or PO, meaning TO = 5, 7, 11 since TP is either 5 or 7. Their sum is 23.
I know why the second answer is incorrect, but mathemathh made a mistake that is very easy to overlook.
If TP = 7, then TO can be 7 or 11 to make the triangle isosceles.
If TP = 5, then TO can 5 or 11
Stop right there, though! Your logic is correct, but TO cannot be 5! Why? Triangles have certain properties that they must adhere to for a triangle to exist. One such theorem is called the triangle inequality theorem. It states that the sum of the lengths of two different sides is greater than the remaining third side.
|\(10>11\)||This is false; 10 is not greater than 11, which means that a triangle with sidelengths 5,5,and 11 cannot exist.|
The other side length possibilities adhere to this restriction, thankfully. Now, add 7 and 11. \(17+11=18\).