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Compare the square of the largest side length to the sum of the squares of the other two side lengths.
Do the Segment Lengths Form a Triangle? Yes.
Is the Triangle Acute, Right, or Obtuse? Acute triangle.
We have been given the following segment lengths. We will investigate whether these segment lengths form a triangle. Then, we will determine whether the triangle is acute, right, or obtuse. 12, 15, and 10sqrt(3) Let's start!
289< 300 <324 ⇕ 17^2< (10 sqrt(3))^2 < 18^2 We know that 10 sqrt(3) is somewhere between 17 and 18. Let's say that 10 sqrt(3) is approximately equal to 17.2 because, in the perfect square form, 300 is closer to 289 rather than 324. 10sqrt(3) &≈ 17.2 Great! Now, we can use the theorem to verify that the segment lengths form a triangle.
| Inequality | Check |
|---|---|
| 12+15 ? > 10sqrt(3) |
27 >≈ 17.2 ✓ |
| 12+10sqrt(3) ? > 15 | ≈ 29.2 > 24 ✓ |
| 10sqrt(3)+15 ? > 12 | ≈ 32.2>32 ✓ |
We can see that segments with lengths of 12, 15, and 10sqrt(3) satisfy the Triangle Inequality Theorem. Therefore, they form a triangle.
We want to determine whether the triangle formed by the given side lengths is acute, right or obtuse. To do so, we will compare the square of the largest side length to the sum of the squares of the other two side lengths. Let a, b, and c be the lengths of the sides, with c being the longest.
| Condition | Type of Triangle |
|---|---|
| a^2+b^2 < c^2 | Obtuse triangle |
| a^2+b^2 = c^2 | Right triangle |
| a^2+b^2 > c^2 | Acute triangle |