If a triangle has a side with length 6 and another side with length 7, which of the following could NOT be the area of the triangle?
a) 1
b) 6
c) 15
d) 21
e) 25
With two given sides a and b, the biggest area of a triangle is when the two sides are at right angles, giving an area of ab/2.
Can you figure out which of the 5 areas is not possible?
To determine which of the given areas could NOT be the area of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
First, let's determine the range of possible values for the base and height of the triangle. We know that one side of the triangle has a length of 6, and another side has a length of 7. For the base of the triangle, it can be any value between 1 (the difference between 6 and 7) and 13 (the sum of 6 and 7).
Next, let's consider the area options:
a) 1. To check if 1 could be the area, we need to find a combination of base and height that satisfy the area formula. Since the area is given by (1/2) * base * height, we need to find two numbers whose product is 2 and whose sum is between 1 and 13. However, it is not possible to find two such numbers that satisfy the conditions. Therefore, 1 could NOT be the area of the triangle.
b) 6. Similar to option a), we need to find two numbers whose product is 12 (since 2 * 6 = 12) and whose sum is between 1 and 13. It is indeed possible to find two numbers (e.g., 2 and 6) that satisfy these conditions. Therefore, 6 could be the area of the triangle.
c) 15. Again, we need to find two numbers whose product is 30 (since 2 * 15 = 30) and whose sum is between 1 and 13. It is possible to find two numbers (e.g., 5 and 6) that satisfy these conditions. Therefore, 15 could be the area of the triangle.
d) 21. Similarly, we need to find two numbers whose product is 42 (since 2 * 21 = 42) and whose sum is between 1 and 13. It is possible to find two numbers (e.g., 7 and 6) that satisfy these conditions. Therefore, 21 could be the area of the triangle.
e) 25. Once again, we need to find two numbers whose product is 50 (since 2 * 25 = 50) and whose sum is between 1 and 13. However, it is not possible to find two such numbers that satisfy the conditions. Therefore, 25 could NOT be the area of the triangle.
In conclusion, the area options that could NOT be the area of the triangle are:
a) 1
e) 25
Therefore, the correct answer is (a) 1 and (e) 25.