If n is an integer and 3 < n < 8, how many different triangles are there with side lengths of 3, 8, and n?
a. one
b. two
c. three
d. four
e. five
To determine the number of different triangles with side lengths of 3, 8, and n, we need to find the possible values of n that satisfy the given conditions.
Given that 3 < n < 8, we know that n can take the integer values 4, 5, 6, or 7.
Let's go through the values of n one by one:
1) If n = 4, the side lengths of the triangle would be 3, 8, and 4. However, this is not a valid triangle because the sum of any two sides of a triangle must be greater than the third side (Triangle Inequality Theorem). In this case, 3 + 4 = 7, which is not greater than 8. Therefore, n = 4 does not form a triangle.
2) If n = 5, the side lengths of the triangle would be 3, 8, and 5. Using the Triangle Inequality Theorem, we can see that 3 + 5 = 8, which is equal to the third side, 8. Therefore, a triangle is possible with n = 5.
3) If n = 6, the side lengths of the triangle would be 3, 8, and 6. Again, using the Triangle Inequality Theorem, we find that 3 + 6 = 9, which is greater than the third side, 8. Therefore, a triangle is possible with n = 6.
4) If n = 7, the side lengths of the triangle would be 3, 8, and 7. Once more, applying the Triangle Inequality Theorem, we get 3 + 7 = 10, which is greater than the third side, 8. Hence, a triangle is possible with n = 7.
So, out of the four values of n (4, 5, 6, and 7), we found three values (5, 6, and 7) that form valid triangles.
Therefore, the answer is c. three.
The other side has to have a length between 8-3 = 5 and 8 + 3 = 11, but not equal to 5 or 11. Since the length must also be an integer (since they say so) the possible side lengths are 6,7,8, 9 and 10.
That makes five possibilities.