The distance between city A and city B is 22 miles. The distance between city [tex]$B$[/tex] and city [tex]$C$[/tex] is 54 miles. The distance between city [tex]$A$[/tex] and city [tex]$C$[/tex] is 51 miles.
What type of triangle is created by the three cities?
A. an acute triangle, because [tex]$22^2+54^2\ \textgreater \ 54^2$[/tex]
B. an acute triangle, because [tex]$22^2+51^2\ \textgreater \ 54^2$[/tex]
C. an obtuse triangle, because [tex]$22^2+54^2\ \textgreater \ 51^2$[/tex]
D. an obtuse triangle, because [tex]$22^2+51^2\ \textgreater \ 54^2$[/tex]
Answer (1)
The problem involves determining the type of triangle given its side lengths. The approach is to calculate the squares of the side lengths and compare the sum of the squares of the two shorter sides with the square of the longest side.
Calculate the squares of the side lengths: 2 2 2 = 484 , 5 4 2 = 2916 , 5 1 2 = 2601 .
Compare 2 2 2 + 5 1 2 with 5 4 2 : 484 + 2601 = 3085 .
Since 2916"> 3085 > 2916 , 54^2"> 2 2 2 + 5 1 2 > 5 4 2 .
Conclude that the triangle is an acute triangle because the sum of the squares of the two shorter sides is greater than the square of the longest side: 54^2}"> an acute triangle, because 2 2 2 + 5 1 2 > 5 4 2 .
Explanation
Problem Analysis Let a = 22 , b = 54 , and c = 51 be the side lengths of the triangle. We need to determine if the triangle is acute, obtuse, or right. To do this, we will compare a 2 + c 2 with b 2 .
Calculate Squares First, calculate the squares of the side lengths: a 2 = 2 2 2 = 484 b 2 = 5 4 2 = 2916 c 2 = 5 1 2 = 2601
Compare and Conclude Now, compare a 2 + c 2 with b 2 :
a 2 + c 2 = 484 + 2601 = 3085 Since 2916"> 3085 > 2916 , we have b^2"> a 2 + c 2 > b 2 .
Final Answer Since b^2"> a 2 + c 2 > b 2 , the triangle is an acute triangle.
Examples
In architecture, determining the type of triangle formed by three points is crucial for structural stability. For instance, if three support beams form an acute triangle, it generally indicates a more stable structure compared to an obtuse triangle, which might require additional reinforcement. Understanding these geometric relationships helps architects design safer and more efficient buildings.
