An Acute Triangle Has Two Sides Measuring 8 Cm And 10 Cm. What Is The Best Representation Of The Possible Range Of Values For The Third Side, S S S ?A. 2 \textless S \textless 18 2 \ \textless \ S \ \textless \ 18 2 \textless S \textless 18 B. S \textless 6 S \ \textless \ 6 S \textless 6 Or $s

The triangle inequality theorem is a fundamental concept in geometry that states the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This theorem is essential in determining the possible range of values for the third side of a triangle when two sides are given.

Applying the Triangle Inequality Theorem

Given an acute triangle with two sides measuring 8 cm and 10 cm, we can use the triangle inequality theorem to determine the possible range of values for the third side, $s$. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Let's consider the following cases:

• $s + 8 > 10$
• $s + 10 > 8$
• $8 + 10 > s$

Simplifying these inequalities, we get:

• $s > 2$
• $s > -2$
• $s < 18$

The first two inequalities are always true, as $s$ is a length and cannot be negative. The third inequality, $s < 18$, is the most restrictive and determines the upper bound of the possible range of values for $s$.

Determining the Lower Bound

To determine the lower bound of the possible range of values for $s$, we need to consider the fact that the triangle is acute. An acute triangle has all angles less than 90 degrees. This means that the third side, $s$, must be greater than the difference between the lengths of the other two sides.

In this case, the difference between the lengths of the other two sides is $10 - 8 = 2$. Therefore, the lower bound of the possible range of values for $s$ is $s > 2$.

Combining the Bounds

Combining the upper and lower bounds, we get the following possible range of values for the third side, $s$:

$2 < s < 18$

This range of values satisfies the triangle inequality theorem and ensures that the triangle is acute.

Comparing with the Options

Comparing the possible range of values for the third side, $s$, with the options provided, we can see that option A is the correct answer.

Conclusion

In conclusion, the best representation of the possible range of values for the third side, $s$, is $2 < s < 18$. This range of values satisfies the triangle inequality theorem and ensures that the triangle is acute.

References

• Triangle Inequality Theorem
• Acute Triangle

Further Reading

• Geometry
• Mathematics

Related Topics

• Triangle Inequality Theorem
• Acute Triangle
• Geometry
• Mathematics
  Frequently Asked Questions (FAQs) =====================================

Q: What is the triangle inequality theorem?

A: The triangle inequality theorem is a fundamental concept in geometry that states the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Q: Why is the triangle inequality theorem important?

A: The triangle inequality theorem is essential in determining the possible range of values for the third side of a triangle when two sides are given. It helps to ensure that the triangle is valid and can be constructed.

Q: What is an acute triangle?

A: An acute triangle is a triangle with all angles less than 90 degrees. This means that the third side, $s$, must be greater than the difference between the lengths of the other two sides.

Q: How do I determine the possible range of values for the third side, $s$?

A: To determine the possible range of values for the third side, $s$, you need to consider the triangle inequality theorem and the fact that the triangle is acute. You can use the following steps:

1. Simplify the inequalities $s + 8 > 10$ and $s + 10 > 8$.
2. Determine the lower bound of the possible range of values for $s$ by considering the difference between the lengths of the other two sides.
3. Combine the upper and lower bounds to get the possible range of values for $s$.

Q: What is the possible range of values for the third side, $s$, in an acute triangle with two sides measuring 8 cm and 10 cm?

A: The possible range of values for the third side, $s$, is $2 < s < 18$.

Q: How do I compare the possible range of values for the third side, $s$, with the options provided?

A: To compare the possible range of values for the third side, $s$, with the options provided, you need to check if the range of values satisfies the triangle inequality theorem and ensures that the triangle is acute.

Q: What is the best representation of the possible range of values for the third side, $s$?

A: The best representation of the possible range of values for the third side, $s$, is $2 < s < 18$.

Q: What are some related topics to the triangle inequality theorem and acute triangles?

A: Some related topics to the triangle inequality theorem and acute triangles include:

• Geometry
• Mathematics
• Triangle Inequality Theorem
• Acute Triangle

Q: Where can I find more information on the triangle inequality theorem and acute triangles?

A: You can find more information on the triangle inequality theorem and acute triangles on the following websites:

• Triangle Inequality Theorem
• Acute Triangle
• Geometry
• Mathematics

Conclusion

In conclusion, the triangle inequality theorem and acute triangles are fundamental concepts in geometry that are essential in determining the possible range of values for the third side of a triangle when two sides are given. By understanding these concepts, you can solve problems and determine the possible range of values for the third side, $s$.