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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? Right 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. 32, 24, and 40 Let's start!
In this step, we will use the Triangle Inequality Theorem to verify that the segment lengths form a triangle.
| Inequality | Check |
|---|---|
| 32+24 ? > 40 |
56 >40 ✓ |
| 32+40 ? > 24 | 72>24 ✓ |
| 40+24 ? > 32 | 64>32 ✓ |
We can see that segments with lengths of 32, 24, and 40 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 |
Let's now consider the given side lengths 24, 32, and 40. Since 40 is the greatest of the numbers, we will let c be 40. We will also arbitrarily let a be 32 and b be 24. 32^2+24^2 ? 40^2 Let's simplify the above statement to determine whether the left-hand side is less than, equal to, or greater than the right-hand side.
Referring back to our table, we can conclude that the side lengths 24, 32, and 40 form a right triangle.