Question

3
select the correct answer from each drop - down menu.
a carpenter is building a triangular frame. she has three pieces of wood measuring 5 feet, 6 feet, and 12 feet.
the carpenter build the triangular frame with the three pieces of wood. she could cut the piece by to create a triangular frame.
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Explanation:

To solve this, we use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Step 1: Check the triangle inequality for the given sides

We have side lengths \( a = 5 \) feet, \( b = 6 \) feet, and \( c = 12 \) feet.

• Check \( a + b \) vs \( c \): \( 5 + 6 = 11 \), and \( 11 < 12 \). So the triangle inequality is not satisfied with these lengths. Thus, the carpenter cannot build the triangular frame with the three pieces as they are.

Step 2: Determine which piece to cut

Since the longest side (12 feet) is the one causing the inequality to fail (because \( 5 + 6 < 12 \)), we need to cut the 12 - foot piece. We need to make it shorter so that \( 5 + 6 > \) (new length of the 12 - foot piece). Let the new length be \( x \). We need \( 5+6>x \), so \( x < 11 \). So we need to cut the 12 - foot piece by at least \( 12 - 10 = 2 \) feet (or any amount such that the new length is less than 11 feet). A common way is to cut it to make the sum of the other two sides greater than the new length. For example, if we cut the 12 - foot piece to 10 feet (cutting it by \( 12 - 10 = 2 \) feet), then \( 5+6 = 11>10 \), \( 5 + 10=15>6 \), and \( 6 + 10 = 16>5 \), which satisfies the triangle inequality.

Final Answers (for the drop - down menus):

1. The first drop - down: cannot
2. The second drop - down: 12 - foot
3. The third drop - down: (an amount such that the new length is less than 11 feet, e.g.,) 2 feet (or other valid amounts like 3 feet, 4 feet, etc., as long as the new length is less than 11)

(Note: The exact amount for the third drop - down can vary as long as it makes the new length of the 12 - foot piece satisfy the triangle inequality with 5 and 6. A typical answer could be cutting the 12 - foot piece by at least 1 foot (but more precisely, at least \( 12 - 10 = 2 \) feet to make it 10 feet, but any amount that makes the new length \( < 11 \) is correct).)

So filling in the blanks:
The carpenter \(\boldsymbol{\text{cannot}}\) build the triangular frame with the three pieces of wood. She could cut the \(\boldsymbol{\text{12 - foot}}\) piece by \(\boldsymbol{\text{2 feet (or more, such that new length}<11\text{ feet)}}\) to create a triangular frame.

Answer:

To solve this, we use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Step 1: Check the triangle inequality for the given sides

We have side lengths \( a = 5 \) feet, \( b = 6 \) feet, and \( c = 12 \) feet.

• Check \( a + b \) vs \( c \): \( 5 + 6 = 11 \), and \( 11 < 12 \). So the triangle inequality is not satisfied with these lengths. Thus, the carpenter cannot build the triangular frame with the three pieces as they are.

Step 2: Determine which piece to cut

Since the longest side (12 feet) is the one causing the inequality to fail (because \( 5 + 6 < 12 \)), we need to cut the 12 - foot piece. We need to make it shorter so that \( 5 + 6 > \) (new length of the 12 - foot piece). Let the new length be \( x \). We need \( 5+6>x \), so \( x < 11 \). So we need to cut the 12 - foot piece by at least \( 12 - 10 = 2 \) feet (or any amount such that the new length is less than 11 feet). A common way is to cut it to make the sum of the other two sides greater than the new length. For example, if we cut the 12 - foot piece to 10 feet (cutting it by \( 12 - 10 = 2 \) feet), then \( 5+6 = 11>10 \), \( 5 + 10=15>6 \), and \( 6 + 10 = 16>5 \), which satisfies the triangle inequality.

Final Answers (for the drop - down menus):

1. The first drop - down: cannot
2. The second drop - down: 12 - foot
3. The third drop - down: (an amount such that the new length is less than 11 feet, e.g.,) 2 feet (or other valid amounts like 3 feet, 4 feet, etc., as long as the new length is less than 11)

(Note: The exact amount for the third drop - down can vary as long as it makes the new length of the 12 - foot piece satisfy the triangle inequality with 5 and 6. A typical answer could be cutting the 12 - foot piece by at least 1 foot (but more precisely, at least \( 12 - 10 = 2 \) feet to make it 10 feet, but any amount that makes the new length \( < 11 \) is correct).)

So filling in the blanks:
The carpenter \(\boldsymbol{\text{cannot}}\) build the triangular frame with the three pieces of wood. She could cut the \(\boldsymbol{\text{12 - foot}}\) piece by \(\boldsymbol{\text{2 feet (or more, such that new length}<11\text{ feet)}}\) to create a triangular frame.