Question
upstudy study bank question image url

(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

error msg
Explain
Simplify this solution

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!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy