The diagram shows three aid stations in a national park. Choose the values of x, y, and z that COULD represent the distances between the stations.
Responses
A x = 8 miles, y = 12 miles, z = 15 miles
B x = 4 miles, y = 7 miles, z = 12 miles
C x = 5 miles, y = 9 miles, z = 14 miles
D x = 7 miles, y = 8 miles, z = 18 miles
To determine which values of x, y, and z could represent the distances between the aid stations, we need to consider 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 check each set of values:
A) x = 8 miles, y = 12 miles, z = 15 miles
For this choice, x + y = 8 + 12 = 20 miles, which is greater than z (15 miles).
Also, x + z = 8 + 15 = 23 miles, which is also greater than y (12 miles).
And, y + z = 12 + 15 = 27 miles, which is greater than x (8 miles).
Therefore, choice A satisfies the triangle inequality theorem.
B) x = 4 miles, y = 7 miles, z = 12 miles
In this case, x + y = 4 + 7 = 11 miles, which is less than z (12 miles).
Therefore, choice B does not satisfy the triangle inequality theorem.
C) x = 5 miles, y = 9 miles, z = 14 miles
For choice C, x + y = 5 + 9 = 14 miles, which is equal to z (14 miles). This is considered a degenerate triangle, where the three sides lie on the same straight line.
Therefore, choice C does not satisfy the triangle inequality theorem.
D) x = 7 miles, y = 8 miles, z = 18 miles
Similar to choice B, x + y = 7 + 8 = 15 miles, which is less than z (18 miles).
Therefore, choice D does not satisfy the triangle inequality theorem.
Based on the triangle inequality theorem, the only set of values that could represent the distances between the aid stations is:
A) x = 8 miles, y = 12 miles, z = 15 miles
The distances between the aid stations in a national park can be represented by the values of x, y, and z. Let's analyze the given options:
A) x = 8 miles, y = 12 miles, z = 15 miles: These values could represent the distances between the stations.
B) x = 4 miles, y = 7 miles, z = 12 miles: These values could represent the distances between the stations.
C) x = 5 miles, y = 9 miles, z = 14 miles: These values could represent the distances between the stations.
D) x = 7 miles, y = 8 miles, z = 18 miles: These values could not represent the distances between the stations because the sum of the two smaller distances (7 miles and 8 miles) is less than the longest distance (18 miles). This violates the triangle inequality theorem and indicates that these distances cannot form a triangle.
Therefore, the values that COULD represent the distances between the stations are:
A) x = 8 miles, y = 12 miles, z = 15 miles
B) x = 4 miles, y = 7 miles, z = 12 miles
C) x = 5 miles, y = 9 miles, z = 14 miles