Question

exercises is it possible to form a triangle with the given side lengths? if not, explain why not. 1. 3, 4, 6 2. 6, 9, 15 3. 8, 8, 8 4. 2, 4, 5 5. 4, 8, 16 6. 1.5, 2.5, 3

Explanation:

Step1: Apply triangle inequality rule

For side lengths $a,b,c$, the sum of any two sides must be greater than the third side: $a+b>c$, $a+c>b$, $b+c>a$.
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For 1. 3, 4, 6

Step1: Check all 3 inequalities

$3+4>6 \implies 7>6$; $3+6>4 \implies 9>4$; $4+6>3 \implies 10>3$
All inequalities hold true.

For 2. 6, 9, 15

Step1: Check key inequality

$6+9=15$, which is not greater than 15.
The triangle inequality fails.

For 3. 8, 8, 8

Step1: Check all 3 inequalities

$8+8>8 \implies 16>8$; this holds for all pairs.
All inequalities hold true.

For 4. 2, 4, 5

Step1: Check all 3 inequalities

$2+4>5 \implies 6>5$; $2+5>4 \implies 7>4$; $4+5>2 \implies 9>2$
All inequalities hold true.

For 5. 4, 8, 16

Step1: Check key inequality

$4+8=12$, which is less than 16.
The triangle inequality fails.

For 6. 1.5, 2.5, 3

Step1: Check all 3 inequalities

$1.5+2.5>3 \implies 4>3$; $1.5+3>2.5 \implies 4.5>2.5$; $2.5+3>1.5 \implies 5.5>1.5$
All inequalities hold true.

Answer:

1. Yes, a triangle can be formed.
2. No, because $6+9=15$, which does not satisfy the triangle inequality (sum of two sides must be greater than the third side).
3. Yes, a triangle can be formed.
4. Yes, a triangle can be formed.
5. No, because $4+8=12<16$, which violates the triangle inequality.
6. Yes, a triangle can be formed.