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?

2 < s < 18
6 < s < 12.8
s < 2 or s > 18
s < 6 or s > 12.8

To find the possible range of values for the third side \(s\) of an acute triangle given two sides, we can use the **triangle inequality theorem** and the **condition for an acute triangle**.

### Step 1: Apply the Triangle Inequality Theorem
For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let the sides of the triangle be \(a = 8 \, \text{cm}\), \(b = 10 \, \text{cm}\), and \(s\) be the unknown side. The inequalities are:

\[
a + b > s \implies 8 + 10 > s \implies s < 18
\]
\[
a + s > b \implies 8 + s > 10 \implies s > 2
\]
\[
b + s > a \implies 10 + s > 8 \implies s > -2 \quad (\text{this inequality is already satisfied for any positive } s)
\]

So from the triangle inequality theorem, we get the range:
\[
2 < s < 18
\]

### Step 2: Ensure the Triangle is Acute
For a triangle to be acute, the square of each side must satisfy the condition:
\[
a^2 + b^2 > s^2
\]
Applying this for our known sides, \(a = 8 \, \text{cm}\) and \(b = 10 \, \text{cm}\):

\[
8^2 + 10^2 > s^2 \implies 64 + 100 > s^2 \implies 164 > s^2 \implies s < \sqrt{164} \approx 12.8
\]

So for the triangle to remain acute, the third side must also satisfy \(s < 12.8\).

### Conclusion
Combining both conditions, we get the range:
\[
6 < s < 12.8
\]
Thus, the correct answer is:
\[
\boxed{6 < s < 12.8}
\]

Do you have any questions or need more details?

Here are 5 related questions:
1. How does the triangle inequality theorem apply to obtuse triangles?
2. Can you explain how to calculate the area of an acute triangle using Heron's formula?
3. What are the properties of an equilateral triangle in relation to the triangle inequality?
4. How do you determine if a triangle is right, acute, or obtuse based on its side lengths?
5. What is the geometric significance of the circumcenter in an acute triangle?

**Tip:** Always double-check the conditions (like the triangle inequality) when working with triangle side lengths to avoid impossible cases.

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?

Explore how to determine the possible range of values for the third side of an acute triangle with sides measuring 8 cm and 10 cm. Learn to apply the triangle inequality theorem and the condition for an acute triangle to find that the range is 6 < s < 12.8. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation