Chapter Test
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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. 4, sqrt()67, and 9 Let's start!
64 < 67 < 81
⇕
8^2< ( sqrt(67))^2 < 9^2
Notice that, in the perfect square form, 67 is closer to 64. Therefore, the approximate value of sqrt(67) can be found as follows.
sqrt(67) &≈ 8.2
Great! Now, we can use the theorem to verify that the segment lengths form a triangle.
| Inequality | Check |
|---|---|
| 4+ sqrt(67) ? > 9 |
≈ 12.2 > 9 ✓ |
| 4+9 ? > sqrt(67) | 13 > ≈ 8.2 ✓ |
| sqrt(67) + 9 ? > 4 | ≈ 17.2 > 4 ✓ |
We can see that the segments with lengths of 4, sqrt(67), and 9 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 |