Question

1. on the triangle below, draw a median and describe what it does to the side.
2. on the triangle below, draw an angle bisector and describe what it does to the angle.
3. if two sides are 8 and 15, what could the length of the third side be?
4. if two sides are 7 and 12, what could the length of the third side be?
5. can 8 ft, 5 ft, 4 ft make a triangle?
6. can 12 ft, 7 ft, 20 ft make a triangle?
7. can 11 ft, 9 ft, 15 ft make a triangle?
8. find the value of x.

(image of triangle for questions 1,2 and a triangle with segments labeled 4x - 5, 2x + 11 for question 8)

Explanation:

Problem 3

Step1: Recall triangle inequality theorem

The triangle inequality theorem states that the length of the third side \( c \) of a triangle with two sides \( a \) and \( b \) must satisfy \( |a - b| < c < a + b \). Here, \( a = 8 \) and \( b = 15 \).

Step2: Calculate the range

First, find the difference: \( 15 - 8 = 7 \). Then, find the sum: \( 15 + 8 = 23 \). So, the third side \( c \) must satisfy \( 7 < c < 23 \). Any number between 7 and 23 (not including 7 and 23) is a possible length, for example, 10.

Step1: Apply triangle inequality theorem

For two sides \( a = 7 \) and \( b = 12 \), the third side \( c \) must satisfy \( |a - b| < c < a + b \).

Step2: Determine the range

Calculate the difference: \( 12 - 7 = 5 \). Calculate the sum: \( 12 + 7 = 19 \). Thus, \( 5 < c < 19 \). A possible length could be 8.

Step1: Check triangle inequalities

For sides 8 ft, 5 ft, and 4 ft, we need to check all three inequalities:

• \( 5 + 4 > 8 \): \( 9 > 8 \) (true)
• \( 5 + 8 > 4 \): \( 13 > 4 \) (true)
• \( 4 + 8 > 5 \): \( 12 > 5 \) (true)

Step2: Conclude

Since all three inequalities hold, the sides can form a triangle.

Answer:

The length of the third side must be greater than 7 and less than 23 (e.g., 10).

Problem 4