Can a triangle have sides with the given lengths? Explain.
15) 8cm, 7cm, 9cm
16) 7ft, 13ft, 6ft
17) 20in, 18in, 16in
18) 3m, 11m, 7m
A triangle can be constructed if any side is shorter the sum of the other two sides.
For example, 4,6,11 cannot be constructed because 4+6 âf; 11,
while 3,4,5 can be constructed because
3+4>5
4+5>3
5+3>4
So check up on the sets, I will do #15:
8+7>9
7+9>8
9+8>7
So answer to #15 is yes.
THANK YOU SO MUCH!
In order for a triangle to exist, the sum of any two sides of a triangle must be greater than the length of the third side. Let's analyze each set of given side lengths to determine if a triangle can be formed.
15) 8cm, 7cm, 9cm: To check if a triangle is possible, we need to test if the sum of any two sides is greater than the third side. In this case, 8cm + 7cm = 15cm, which is greater than 9cm. Also, 8cm + 9cm = 17cm, which is greater than 7cm. Finally, 7cm + 9cm = 16cm, which is greater than 8cm. Thus, a triangle can be formed with these side lengths.
16) 7ft, 13ft, 6ft: Similarly, we need to check if the sum of any two sides is greater than the third side. Here, 7ft + 13ft = 20ft, which is greater than 6ft. Also, 7ft + 6ft = 13ft, which is equal to the third side. Finally, 13ft + 6ft = 19ft, which is greater than 7ft. So, a triangle is possible.
17) 20in, 18in, 16in: Following the same logic, we check if the sum of any two sides is greater than the third side. Here, 20in + 18in = 38in, which is greater than 16in. Also, 20in + 16in = 36in, which is greater than 18in. Similarly, 18in + 16in = 34in, which is greater than 20in. So, a triangle is possible with these side lengths.
18) 3m, 11m, 7m: Let's apply the same concept and check if the sum of any two sides is greater than the third side. Here, 3m + 11m = 14m, which is less than 7m. Thus, the triangle with these side lengths is not possible.
To summarize:
- A triangle can be formed with side lengths of 8cm, 7cm, and 9cm.
- A triangle can be formed with side lengths of 7ft, 13ft, and 6ft.
- A triangle can be formed with side lengths of 20in, 18in, and 16in.
- A triangle cannot be formed with side lengths of 3m, 11m, and 7m.
To determine if a triangle can have sides with the given lengths, we can apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's evaluate each set of side lengths one by one:
15) 8cm, 7cm, 9cm.
To check if this can form a triangle, we need to test if the sum of the two smaller sides is greater than the length of the largest side:
8 + 7 = 15 (sum of the two smaller sides)
15 > 9 (length of the largest side)
Since the sum of the two smaller sides is greater than the length of the largest side, this set of side lengths can form a triangle.
16) 7ft, 13ft, 6ft.
Let's test the sum of the two smaller sides:
7 + 6 = 13
13 is equal to the length of the third side (13ft).
Since the sum of the two smaller sides is equal to the length of the third side, this set of side lengths can form a degenerate triangle (a triangle with all three sides lying on the same line).
17) 20in, 18in, 16in.
Sum of the two smaller sides:
18 + 16 = 34
34 is greater than the length of the largest side (20in).
Since the sum of the two smaller sides is greater than the length of the largest side, this set of side lengths can form a triangle.
18) 3m, 11m, 7m.
Sum of the two smaller sides:
3 + 7 = 10
10 is less than the length of the largest side (11m).
Since the sum of the two smaller sides is less than the length of the largest side, this set of side lengths cannot form a triangle.
In summary:
- The set of side lengths (8cm, 7cm, 9cm) can form a triangle.
- The set of side lengths (7ft, 13ft, 6ft) can form a degenerate triangle.
- The set of side lengths (20in, 18in, 16in) can form a triangle.
- The set of side lengths (3m, 11m, 7m) cannot form a triangle.