Question

1. 19, 7, 10

two sides of a triangle have the following measures. find the range of possible measures for
the third side.

1. 9, 12 3 < x < 21
2. 9, 7 2
3. 10, 6 4
4. 11, 8 3
5. 7, 12 5
6. 10, 11 1

Explanation:

Step1: Recall triangle inequality rule

For a triangle with sides $a, b, c$, the sum of any two sides must be greater than the third side. For two given sides, let the third side be $x$: $|a-b| < x < a+b$.

Step2: Solve for 9,7

Find difference and sum: $|9-7|=2$, $9+7=16$
Expression: $2 < x < 16$

Step3: Solve for 10,6

Find difference and sum: $|10-6|=4$, $10+6=16$
Expression: $4 < x < 16$

Step4: Solve for 11,8

Find difference and sum: $|11-8|=3$, $11+8=19$
Expression: $3 < x < 19$

Step5: Solve for 7,12

Find difference and sum: $|12-7|=5$, $12+7=19$
Expression: $5 < x < 19$

Step6: Solve for 10,11

Find difference and sum: $|11-10|=1$, $11+10=21$
Expression: $1 < x < 21$

Answer:

1. $3 < x < 21$
2. $2 < x < 16$
3. $4 < x < 16$
4. $3 < x < 19$
5. $5 < x < 19$
6. $1 < x < 21$