Question
(a) One triangle has side lengths 12,14 , and 10 . Compute the sum of the squares of the shorter lengths \( 10^{2}+12^{2}=\square \) Compute the square of the longest length. \( 14^{2}=\square \) What kind of triangle is it? Acute triangle Right triangle Obtuse triangle
Ask by Hill Boone. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The triangle has side lengths 10, 12, and 14.
Sum of the squares of the shorter lengths: \( 10^{2} + 12^{2} = 244 \)
Square of the longest length: \( 14^{2} = 196 \)
Since \( 244 > 196 \), the triangle is an **acute triangle**.
Solution
Let's analyze the given triangle with side lengths 10, 12, and 14.
### 1. Sum of the Squares of the Shorter Lengths
Compute \( 10^{2} + 12^{2} \):
\[
10^{2} + 12^{2} = 100 + 144 = 244
\]
### 2. Square of the Longest Length
Compute \( 14^{2} \):
\[
14^{2} = 196
\]
### 3. Determining the Type of Triangle
To determine the type of triangle, compare the sum of the squares of the two shorter sides with the square of the longest side:
- If \( a^{2} + b^{2} > c^{2} \), the triangle is **acute**.
- If \( a^{2} + b^{2} = c^{2} \), the triangle is **right**.
- If \( a^{2} + b^{2} < c^{2} \), the triangle is **obtuse**.
In this case:
\[
244 > 196
\]
Since \( 10^{2} + 12^{2} > 14^{2} \), the triangle is an **acute triangle**.
**Answer:**
\[
10^{2} + 12^{2} = 244
\]
\[
14^{2} = 196
\]
**The triangle is an acute triangle.**
Reviewed and approved by the UpStudy tutoring team
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The Deep Dive
Let's do some quick calculations! The sum of the squares of the shorter lengths is \( 10^2 + 12^2 = 100 + 144 = 244 \). Now, for the longest length, we calculate \( 14^2 = 196 \). To determine what kind of triangle this is, we can use the relationship between the squares of the sides. Since \( 244 > 196 \), this triangle is classified as an obtuse triangle—where one angle is greater than 90 degrees. How cool is that? Triangles sure can have personality!