Which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.?
A. acute, because [tex]$10^2+12^2\ \textgreater \ 15^2$[/tex]
B. acute, because [tex]$12^2+15^2\ \textgreater \ 10^2$[/tex]
C. obtuse, because [tex]$10^2+12^2\ \textgreater \ 15^2$[/tex]
D. obtuse, because [tex]$12^2+15^2\ \textgreater \ 10^2$[/tex]
Answer (2)
Calculate the sum of the squares of the two shorter sides: 1 0 2 + 1 2 2 = 244 .
Calculate the square of the longest side: 1 5 2 = 225 .
Compare the two values: 225"> 244 > 225 , which means 15^2"> 1 0 2 + 1 2 2 > 1 5 2 .
Since the sum of the squares of the two shorter sides is greater than the square of the longest side, the triangle is acute. 15^2}"> acute, because 1 0 2 + 1 2 2 > 1 5 2
Explanation
Analyze the problem and given data We are given a triangle with side lengths 10 inches, 12 inches, and 15 inches. We need to classify this triangle as acute, obtuse, or right. To do this, we will compare the square of the longest side with the sum of the squares of the other two sides.
The longest side is 15 inches. The other two sides are 10 inches and 12 inches.
Calculate the squares of the sides First, let's calculate the sum of the squares of the two shorter sides: 1 0 2 + 1 2 2 = 100 + 144 = 244
Now, let's calculate the square of the longest side: 1 5 2 = 225
Compare the calculated values Now, we compare the two values: 1 0 2 + 1 2 2 = 244 1 5 2 = 225
Since 225"> 244 > 225 , we have 15^2"> 1 0 2 + 1 2 2 > 1 5 2 .
Classify the triangle If the sum of the squares of the two shorter sides is greater than the square of the longest side, then the triangle is acute. In our case, 15^2"> 1 0 2 + 1 2 2 > 1 5 2 , so the triangle is acute.
Therefore, the triangle is acute because 15^2"> 1 0 2 + 1 2 2 > 1 5 2 .
Examples
Understanding triangle classifications is useful in architecture and engineering. For example, when designing a roof, knowing whether a triangle is acute, right, or obtuse helps determine the angles and structural support needed. An acute triangle might be used for a gently sloping roof, while an obtuse triangle might be avoided due to its wider angle requiring more material and support. Correctly classifying triangles ensures stable and efficient designs.
The triangle with side lengths 10 in., 12 in., and 15 in. is classified as acute because the sum of the squares of the two shorter sides (244) is greater than the square of the longest side (225). Therefore, the chosen option is A. acute, because 15^2"> 1 0 2 + 1 2 2 > 1 5 2 .
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