Question

two sides of $\triangle abc$ have lengths of 4 inches and 7 inches.\
a. what are the possible lengths of the third side?\
b. what are the possible whole - number lengths of the third side?\

1. two sides of $\triangle xyz$ have lengths of 10 cm and 14 cm.\

a. what are the possible lengths of the third side?\
b. what are the possible whole - number lengths of the third side?\
remember\
for problems 9 - 12, find the percent of the number.\

1. 10% of 50\
2. 25% of 60\
3. 30% of 60\
4. 75% of 80

Explanation:

Step1: Apply triangle inequality theorem

Let the third side of $\triangle ABC$ be $x$. The triangle inequality states that the sum of any two sides must be greater than the third side. So we have three inequalities:

1. $4 + 7 > x$ → $11 > x$
2. $4 + x > 7$ → $x > 7 - 4$ → $x > 3$
3. $7 + x > 4$ → $x > 4 - 7$ → $x > -3$ (this is always true for positive lengths, so we ignore it)

Step2: Combine valid inequalities

The valid range is $3 < x < 11$

Step3: Find whole numbers in the range

Whole numbers between 3 and 11 (not including 3 and 11) are 4, 5, 6, 7, 8, 9, 10

Step4: Apply theorem to $\triangle XYZ$

Let the third side of $\triangle XYZ$ be $y$. Use triangle inequality:

1. $10 + 14 > y$ → $24 > y$
2. $10 + y > 14$ → $y > 14 - 10$ → $y > 4$
3. $14 + y > 10$ → $y > 10 - 14$ → $y > -4$ (ignore, since length is positive)

Step5: Combine valid inequalities for $\triangle XYZ$

The valid range is $4 < y < 24$

Step6: Find whole numbers in the range

Whole numbers between 4 and 24 (not including 4 and 24) are 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

Step7: Calculate 10% of 50

$10\% \times 50 = \frac{10}{100} \times 50 = 5$

Step8: Calculate 25% of 60

$25\% \times 60 = \frac{25}{100} \times 60 = 15$

Step9: Calculate 30% of 60

$30\% \times 60 = \frac{30}{100} \times 60 = 18$

Step10: Calculate 75% of 80

$75\% \times 80 = \frac{75}{100} \times 80 = 60$

Answer:

1. For $\triangle ABC$:

a. $3 < x < 11$ inches
b. 4, 5, 6, 7, 8, 9, 10 inches

1. For $\triangle XYZ$:

a. $4 < y < 24$ cm
b. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 cm

1. Percent problems:
2. 5
3. 15
4. 18
5. 60