Solved on Jan 22, 2024

Identify the set of 3 numbers that could represent the sides of a triangle. Options: $\{4,16,21\}$ , $\{9,14,20\}$ , $\{11,14,27\}$ , $\{6,14,20\}$ .

STEP 1

Assumptions
1. We are given four sets of three numbers each.
2. We need to determine which set(s) can represent the sides of a triangle.
3. To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem.

STEP 2

We will apply the triangle inequality theorem to each set of numbers. The triangle inequality theorem states that for any triangle with sides of lengths $a$ , $b$ , and $c$ , the following must be true:
$a + b > c$ $a + c > b$ $b + c > a$

STEP 3

Check the first set $\{4,16,21\}$ using the triangle inequality theorem.

STEP 4

Check if $4 + 16 > 21$ .
$4 + 16 = 20$
Since $20$ is not greater than $21$ , the first set does not satisfy the triangle inequality theorem.

STEP 5

Check the second set $\{9,14,20\}$ using the triangle inequality theorem.

STEP 6

Check if $9 + 14 > 20$ .
$9 + 14 = 23$
Since $23$ is greater than $20$ , the first condition is satisfied.

STEP 7

Check if $9 + 20 > 14$ .
$9 + 20 = 29$
Since $29$ is greater than $14$ , the second condition is satisfied.

STEP 8

Check if $14 + 20 > 9$ .
$14 + 20 = 34$
Since $34$ is greater than $9$ , the third condition is satisfied.

STEP 9

Since all three conditions of the triangle inequality theorem are satisfied, the second set $\{9,14,20\}$ can represent the sides of a triangle.

STEP 10

Check the third set $\{11,14,27\}$ using the triangle inequality theorem.

STEP 11

Check if $11 + 14 > 27$ .
$11 + 14 = 25$
Since $25$ is not greater than $27$ , the third set does not satisfy the triangle inequality theorem.

STEP 12

Check the fourth set $\{6,14,20\}$ using the triangle inequality theorem.

STEP 13

Check if $6 + 14 > 20$ .
$6 + 14 = 20$
Since $20$ is not greater than $20$ , the fourth set does not satisfy the triangle inequality theorem.

STEP 14

Conclude which set(s) can represent the sides of a triangle.
The only set that satisfies the triangle inequality theorem is $\{9,14,20\}$ .
Therefore, the set of numbers that could represent the three sides of a triangle is $\{9,14,20\}$ .

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Identify the set of 3 numbers that could represent the sides of a triangle. Options: $\{4,16,21\}$ , $\{9,14,20\}$ , $\{11,14,27\}$ , $\{6,14,20\}$ .

STEP 1

Assumptions
1. We are given four sets of three numbers each.
2. We need to determine which set(s) can represent the sides of a triangle.
3. To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem.

STEP 2

We will apply the triangle inequality theorem to each set of numbers. The triangle inequality theorem states that for any triangle with sides of lengths $a$ , $b$ , and $c$ , the following must be true:
$a + b > c$ $a + c > b$ $b + c > a$

STEP 3

Check the first set $\{4,16,21\}$ using the triangle inequality theorem.

STEP 4

Check if $4 + 16 > 21$ .
$4 + 16 = 20$
Since $20$ is not greater than $21$ , the first set does not satisfy the triangle inequality theorem.

STEP 5

Check the second set $\{9,14,20\}$ using the triangle inequality theorem.

STEP 6

Check if $9 + 14 > 20$ .
$9 + 14 = 23$
Since $23$ is greater than $20$ , the first condition is satisfied.

STEP 7

Check if $9 + 20 > 14$ .
$9 + 20 = 29$
Since $29$ is greater than $14$ , the second condition is satisfied.

STEP 8

Check if $14 + 20 > 9$ .
$14 + 20 = 34$
Since $34$ is greater than $9$ , the third condition is satisfied.

STEP 9

Since all three conditions of the triangle inequality theorem are satisfied, the second set $\{9,14,20\}$ can represent the sides of a triangle.

STEP 10

Check the third set $\{11,14,27\}$ using the triangle inequality theorem.

STEP 11

Check if $11 + 14 > 27$ .
$11 + 14 = 25$
Since $25$ is not greater than $27$ , the third set does not satisfy the triangle inequality theorem.

STEP 12

Check the fourth set $\{6,14,20\}$ using the triangle inequality theorem.

STEP 13

Check if $6 + 14 > 20$ .
$6 + 14 = 20$
Since $20$ is not greater than $20$ , the fourth set does not satisfy the triangle inequality theorem.

STEP 14

Conclude which set(s) can represent the sides of a triangle.
The only set that satisfies the triangle inequality theorem is $\{9,14,20\}$ .
Therefore, the set of numbers that could represent the three sides of a triangle is $\{9,14,20\}$ .

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Identify the set of 3 numbers that could represent the sides of a triangle. Options: {4,16,21}\{4,16,21\}{4,16,21}, {9,14,20}\{9,14,20\}{9,14,20}, {11,14,27}\{11,14,27\}{11,14,27}, {6,14,20}\{6,14,20\}{6,14,20}.

STEP 1

Assumptions1. We are given four sets of three numbers each.2. We need to determine which set(s) can represent the sides of a triangle.3. To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem.

STEP 2

We will apply the triangle inequality theorem to each set of numbers. The triangle inequality theorem states that for any triangle with sides of lengths aaa, bbb, and ccc, the following must be true:a+b>ca + b > ca+b>c a+c>ba + c > ba+c>b b+c>ab + c > ab+c>a

STEP 3

Check the first set {4,16,21}\{4,16,21\}{4,16,21} using the triangle inequality theorem.

STEP 4

Check if 4+16>214 + 16 > 214+16>21.4+16=204 + 16 = 204+16=20Since 202020 is not greater than 212121, the first set does not satisfy the triangle inequality theorem.

STEP 5

Check the second set {9,14,20}\{9,14,20\}{9,14,20} using the triangle inequality theorem.

STEP 6

Check if 9+14>209 + 14 > 209+14>20.9+14=239 + 14 = 239+14=23Since 232323 is greater than 202020, the first condition is satisfied.

STEP 7

Check if 9+20>149 + 20 > 149+20>14.9+20=299 + 20 = 299+20=29Since 292929 is greater than 141414, the second condition is satisfied.

STEP 8

Check if 14+20>914 + 20 > 914+20>9.14+20=3414 + 20 = 3414+20=34Since 343434 is greater than 999, the third condition is satisfied.

STEP 9

Since all three conditions of the triangle inequality theorem are satisfied, the second set {9,14,20}\{9,14,20\}{9,14,20} can represent the sides of a triangle.

STEP 10

Check the third set {11,14,27}\{11,14,27\}{11,14,27} using the triangle inequality theorem.

STEP 11

Check if 11+14>2711 + 14 > 2711+14>27.11+14=2511 + 14 = 2511+14=25Since 252525 is not greater than 272727, the third set does not satisfy the triangle inequality theorem.

STEP 12

Check the fourth set {6,14,20}\{6,14,20\}{6,14,20} using the triangle inequality theorem.

STEP 13

Check if 6+14>206 + 14 > 206+14>20.6+14=206 + 14 = 206+14=20Since 202020 is not greater than 202020, the fourth set does not satisfy the triangle inequality theorem.

STEP 14

Conclude which set(s) can represent the sides of a triangle.The only set that satisfies the triangle inequality theorem is {9,14,20}\{9,14,20\}{9,14,20}.Therefore, the set of numbers that could represent the three sides of a triangle is {9,14,20}\{9,14,20\}{9,14,20}.

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Identify the set of 3 numbers that could represent the sides of a triangle. Options: $\{4,16,21\}$ , $\{9,14,20\}$ , $\{11,14,27\}$ , $\{6,14,20\}$ .

We will apply the triangle inequality theorem to each set of numbers. The triangle inequality theorem states that for any triangle with sides of lengths $a$ , $b$ , and $c$ , the following must be true:
$a + b > c$ $a + c > b$ $b + c > a$

Check the first set $\{4,16,21\}$ using the triangle inequality theorem.

Check if $4 + 16 > 21$ .
$4 + 16 = 20$
Since $20$ is not greater than $21$ , the first set does not satisfy the triangle inequality theorem.

Check the second set $\{9,14,20\}$ using the triangle inequality theorem.

Check if $9 + 14 > 20$ .
$9 + 14 = 23$
Since $23$ is greater than $20$ , the first condition is satisfied.

Check if $9 + 20 > 14$ .
$9 + 20 = 29$
Since $29$ is greater than $14$ , the second condition is satisfied.

Check if $14 + 20 > 9$ .
$14 + 20 = 34$
Since $34$ is greater than $9$ , the third condition is satisfied.

Since all three conditions of the triangle inequality theorem are satisfied, the second set $\{9,14,20\}$ can represent the sides of a triangle.

Check the third set $\{11,14,27\}$ using the triangle inequality theorem.

Check if $11 + 14 > 27$ .
$11 + 14 = 25$
Since $25$ is not greater than $27$ , the third set does not satisfy the triangle inequality theorem.

Check the fourth set $\{6,14,20\}$ using the triangle inequality theorem.

Check if $6 + 14 > 20$ .
$6 + 14 = 20$
Since $20$ is not greater than $20$ , the fourth set does not satisfy the triangle inequality theorem.

Conclude which set(s) can represent the sides of a triangle.
The only set that satisfies the triangle inequality theorem is $\{9,14,20\}$ .
Therefore, the set of numbers that could represent the three sides of a triangle is $\{9,14,20\}$ .