Question

a triangle with these listed lengths.
a. 8, 12, x
i.
b. 10, 6, x
i.
c. 21, 11, x
i.
d. 27, 15, x
i.
notes:

Explanation:

To determine the possible values of \( x \) for a triangle with given side lengths, we use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For sides \( a \), \( b \), and \( x \), this gives three inequalities: \( a + b > x \), \( a + x > b \), and \( b + x > a \). Simplifying these, we find that \( |a - b| < x < a + b \).

Part a: Sides 8, 12, \( x \)

Step 1: Calculate the difference and sum

First, find the absolute difference of the two given sides: \( |12 - 8| = 4 \). Then, find the sum of the two given sides: \( 12 + 8 = 20 \).

Step 2: Apply the Triangle Inequality

Using the theorem, \( x \) must satisfy \( 4 < x < 20 \).

Part b: Sides 10, 6, \( x \)

Step 1: Calculate the difference and sum

Find the absolute difference: \( |10 - 6| = 4 \). Find the sum: \( 10 + 6 = 16 \).

Step 2: Apply the Triangle Inequality

Thus, \( x \) must satisfy \( 4 < x < 16 \).

Part c: Sides 21, 11, \( x \)

Step 1: Calculate the difference and sum

Find the absolute difference: \( |21 - 11| = 10 \). Find the sum: \( 21 + 11 = 32 \).

Step 2: Apply the Triangle Inequality

So, \( x \) must satisfy \( 10 < x < 32 \).

Part d: Sides 27, 15, \( x \)

Step 1: Calculate the difference and sum

Find the absolute difference: \( |27 - 15| = 12 \). Find the sum: \( 27 + 15 = 42 \).

Step 2: Apply the Triangle Inequality

Therefore, \( x \) must satisfy \( 12 < x < 42 \).

Answer:

• a. \( 4 < x < 20 \)
• b. \( 4 < x < 16 \)
• c. \( 10 < x < 32 \)
• d. \( 12 < x < 42 \)