Question

1.6 triangle inequality
maria draw a triangle with side lengths 3 and 7. which of the following side lengths is a possible third side of the triangle?
a. 4
b. 8
c. 10
d. 12

select the following side lengths that can create a triangle:
□ 3 , 4 , 9
□ 17,17,12
□ 10, 15, 12
□ 19, 21, 13
□ 12, 10, 23
□ 13, 24, 10

Explanation:

First Question (Possible third side)

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for a triangle with sides $a$, $b$, $c$, the sum of any two sides must be greater than the third side. For given sides 3 and 7, let the third side be $x$.
First inequality: $3 + 7 > x$ → $10 > x$
Second inequality: $3 + x > 7$ → $x > 7 - 3$ → $x > 4$
Third inequality: $7 + x > 3$ → $x > 3 - 7$ (this is always true for positive side lengths)
So the range is $4 < x < 10$.

Step2: Match with options

Check which option falls between 4 and 10.

Step1: Test each set with triangle inequality

For each set, verify that the sum of the two shorter sides is greater than the longest side.

1. Set 3,4,9: $3+4=7 < 9$ → Invalid
2. Set 17,17,12: $12+17=29 > 17$ → Valid
3. Set 10,15,12: $10+12=22 > 15$ → Valid
4. Set 19,21,13: $13+19=32 > 21$ → Valid
5. Set 12,10,23: $12+10=22 < 23$ → Invalid
6. Set 13,24,10: $13+10=23 < 24$ → Invalid

Answer:

B. 8

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Second Question (Valid triangle side lengths)