Which set of numbers can represent the side lengths, in inches, of an acute triangle?
A. 4, 5, 7
B. 5, 7, 8
C. 6, 7, 10
D. 7, 9, 12
Answer (1)
Check if each set of numbers satisfies the triangle inequality theorem.
For sets that satisfy the triangle inequality, check if c^2"> a 2 + b 2 > c 2 , where c is the longest side.
If c^2"> a 2 + b 2 > c 2 , then the triangle is acute.
The set 5, 7, 8 satisfies both conditions, so it represents the side lengths of an acute triangle. 5 , 7 , 8
Explanation
Problem Analysis We are given four sets of numbers and we need to determine which one can represent the side lengths of an acute triangle. An acute triangle is a triangle in which all three angles are less than 90 degrees.
Conditions for Acute Triangle For a triangle with side lengths a , b , and c , where c is the longest side, the triangle is acute if c^2"> a 2 + b 2 > c 2 . We also need to check the triangle inequality theorem, which states that the sum of any two sides must be greater than the third side.
Checking Each Set Let's analyze each set:
Set 1: 4, 5, 7
Check triangle inequality: 7"> 4 + 5 > 7 , 5"> 4 + 7 > 5 , 4"> 5 + 7 > 4 . All conditions are met.
Check if 7^2"> 4 2 + 5 2 > 7 2 : 49"> 16 + 25 > 49 , 49"> 41 > 49 . This is false. Therefore, this is not an acute triangle.
Set 2: 5, 7, 8
Check triangle inequality: 8"> 5 + 7 > 8 , 7"> 5 + 8 > 7 , 5"> 7 + 8 > 5 . All conditions are met.
Check if 8^2"> 5 2 + 7 2 > 8 2 : 64"> 25 + 49 > 64 , 64"> 74 > 64 . This is true. Therefore, this is an acute triangle.
Set 3: 6, 7, 10
Check triangle inequality: 10"> 6 + 7 > 10 , 7"> 6 + 10 > 7 , 6"> 7 + 10 > 6 . All conditions are met.
Check if 10^2"> 6 2 + 7 2 > 1 0 2 : 100"> 36 + 49 > 100 , 100"> 85 > 100 . This is false. Therefore, this is not an acute triangle.
Set 4: 7, 9, 12
Check triangle inequality: 12"> 7 + 9 > 12 , 9"> 7 + 12 > 9 , 7"> 9 + 12 > 7 . All conditions are met.
Check if 12^2"> 7 2 + 9 2 > 1 2 2 : 144"> 49 + 81 > 144 , 144"> 130 > 144 . This is false. Therefore, this is not an acute triangle.
Final Answer The only set that satisfies the conditions for an acute triangle is 5, 7, 8.
Examples
Understanding acute triangles is crucial in architecture and engineering. For instance, when designing a roof, ensuring the triangle formed by the rafters is acute helps distribute weight evenly and enhances structural stability. The relationship c^2"> a 2 + b 2 > c 2 guarantees that the angles are less than 90 degrees, preventing excessive stress on joints and materials. This principle ensures the roof can withstand environmental pressures like snow and wind, making buildings safer and more durable.
