Can the Side Lengths 8, 9, and 12 Form a Triangle?

Yes, the side lengths of 8, 9, and 12 can form a valid triangle. A cornerstone of geometry is the triangle inequality theorem, which states that for any three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Verification Through Triangle Inequality Theorem

To verify this, let's apply the triangle inequality theorem:

8 9 > 12 9 12 > 8 12 8 > 9

Each of these conditions is met, confirming that the side lengths 8, 9, and 12 can indeed form a triangle. This is not only a fundamental property but also a fascinating aspect of Euclidean geometry.

Exploring the Triangle Properties

Just for fun, let's delve deeper into what kind of triangle can be formed with these side lengths. We can construct a triangle with side lengths of 8, 9, and 12, and we can determine some of its properties. For instance, we can find the exact height of the triangle when it is standing on the side with length 8 units. This requires a bit of algebraic manipulation and the Pythagorean theorem.

Calculating the Height

Assume that the triangle stands on the side with length 8 units, with the red side being 10 units and the blue side being 11 units. We can use the Pythagorean theorem to solve for the height:

[x^2 h^2 10^2 100]

[8 - x^2 h^2 11^2 121]

Solving for h2 from the first equation:

[h^2 100 - x^2]

Solving for h2 from the second equation:

[h^2 121 - (8 - x^2)]

Setting the two expressions for h2 equal to each other:

[100 - x^2 121 - 8 x^2]

[100 - x^2 113 - x^2]

[16x 113 - 100]

[16x 113 - 100]

[16x 13]

[x 13 / 16]

The height is then:

[h^2 100 - x^2 100 - (13/16)^2 100 - 169/256 25600/256 - 169/256 23931/256]

[h sqrt{23931}/16]

Thus, the height of the triangle is approximately:

[h sqrt{23931}/16 approx 3.16 text{ units}]

Similar Examples

Let's consider another example: the side lengths 8, 11, and 12 also form a triangle according to the same principle. The triangle inequality theorem is satisfied:

8 11 > 12 11 12 > 8 12 8 > 11

Since the condition is met, a triangle with side lengths 8, 11, and 12 is possible.

Law of Cosines

The side lengths of 8, 11, and 12 can also correspond to the Law of Cosines:

c2 a2 b2 - 2ab cos(θ)

For the side lengths 12, 11, and 8, we have:

[12^2 11^2 8^2 - 2 times 11 times 8 times cos(theta)]

[144 121 64 - 176 cos(theta)]

[144 185 - 176 cos(theta)]

[cos(theta) frac{185 - 144}{176} frac{41}{176} approx 0.233]

Since the value of cosine is within the range [-1, 1], this confirms that the set of side lengths 8, 11, and 12 can indeed form a triangle.

Conclusion

In summary, the side lengths 8, 9, and 12 can form a triangle, as can 8, 11, and 12. These examples demonstrate the practical application of the triangle inequality theorem and the Law of Cosines in determining the validity of side lengths in forming triangles. Understanding these principles is crucial for anyone interested in geometry or related fields.