Question

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
\boxed{cannot}
build the triangular frame with the three pieces of wood. she could cut the
\boxed{12 - foot}
piece by
\boxed{11 feet}
\boxed{1 foot}
\boxed{2 feet}
reset
next

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be greater than the length of the remaining side. Mathematically, this means:
\[ a + b > c, \quad a + c > b, \quad b + c > a \]

Step2: Check the original side lengths

The original side lengths are \(5\) feet, \(6\) feet, and \(12\) feet. Let's apply the triangle inequality theorem:
- \(5 + 6 = 11\), and \(11 ot> 12\) (since \(11 < 12\)). So, the original three sides do not satisfy the triangle inequality, meaning the carpenter **cannot** build the triangular frame with the original pieces.

Step3: Determine which piece to cut

We need to cut the \(12\)-foot piece (since the other two sides are \(5\) and \(6\)) such that when we replace \(12\) with the new length \(x\), the triangle inequalities hold. Let's denote the new length of the cut piece as \(x\). The sides will be \(5\), \(6\), and \(x\). The triangle inequalities give:
- \(5 + 6 > x \implies 11 > x\)
- \(5 + x > 6 \implies x > 1\)
- \(6 + x > 5 \implies x > -1\) (which is always true since length is positive)

So, \(x\) must be greater than \(1\) and less than \(11\). Let's check the options for cutting the \(12\)-foot piece:
- If we cut \(11\) feet from the \(12\)-foot piece, the new length is \(12 - 11 = 1\) foot. But \(1\) is not greater than \(1\) (it's equal), so that doesn't work. Wait, no—wait, actually, we are cutting the \(12\)-foot piece to a length \(x\) such that \(x\) is less than \(11\) and greater than \(1\). Wait, maybe I misread the options. Wait, the options for cutting the \(12\)-foot piece are \(11\) feet, \(1\) foot, or \(2\) feet? Wait, no—the drop-down after "cut the 12-foot piece by"—wait, maybe it's "cut the 12-foot piece to" a certain length? Wait, the original problem's drop-down options (from the image) are "11 feet", "1 foot", "2 feet" (as per the visible text: "11 feet", "1 foot", "2 feet" in the drop-down). Wait, let's re-express.

Wait, the carpenter has to cut the \(12\)-foot piece to a length \(x\) so that \(5\), \(6\), and \(x\) form a triangle. Let's check each option for \(x\) (the new length of the \(12\)-foot piece):

- Option 1: \(x = 11\) feet. Check inequalities:
- \(5 + 6 = 11\), which is not greater than \(11\) (it's equal). So, \(5 + 6 ot> 11\), so this doesn't work. Wait, that can't be. Wait, maybe I made a mistake. Wait, no—wait, the triangle inequality is "greater than", not "greater than or equal to". So \(a + b > c\), not \(a + b \geq c\). So, if \(x = 11\), then \(5 + 6 = 11\), which is not greater than \(11\), so that's invalid.

- Option 2: \(x = 1\) foot. Check inequalities:
- \(5 + 1 = 6\), which is not greater than \(6\) (it's equal). So \(5 + 1 ot> 6\), invalid.

- Option 3: \(x = 2\) feet. Check inequalities:
- \(5 + 6 = 11 > 2\)
- \(5 + 2 = 7 > 6\)
- \(6 + 2 = 8 > 5\)

All inequalities are satisfied. Wait, but the drop-down options after "cut the 12-foot piece by"—wait, maybe the options are how much to cut off, not the new length? Wait, the text says "cut the 12-foot piece by"—maybe "cut the 12-foot piece to"—the image shows "11 feet", "1 foot", "2 feet" as options. Wait, maybe the correct approach is:

Wait, the original sides are \(5\), \(6\), \(12\). Since \(5 + 6 < 12\), we need to reduce the length of the \(12\)-foot piece so that \(5 + 6 > \text{new length}\). So \(5 + 6 = 11\), so the new length must be less than \(11\). So we need to cut the \(12\)-foot piece to a length less than \(11\). Let's check the options for the new length:

- If we cut the \(12\)-foot piece to \(11\) feet (cut off \(1\) foot), then the sides are \(5\), \(6\), \(11\). But \(5 + 6 = 11\), which is not greater than \(11\), so that's invalid (degenerate triangle, not a valid triangle).

- If we cut the \(12\)-foot piece to \(2\) feet (cut off \(10\) feet), then the sides are \(5\), \(6\), \(2\). Check:
- \(5 + 6 > 2\) (11 > 2: true)
- \(5 + 2 > 6\) (7 > 6: true)
- \(6 + 2 > 5\) (8 > 5: true)

This works. Wait, but the options are "11 feet", "1 foot", "2 feet" as the new length? Wait, the image shows the drop-down with "11 feet", "1 foot", "2 feet". So the correct new length is \(2\) feet? Wait, no—wait, maybe I messed up the first part. Wait, the first drop-down is "cannot" (since \(5 + 6 < 12\)), and the second drop-down is which piece to cut (the \(12\)-foot piece), and then by how much? Wait, no—the text says "She could cut the 12-foot piece by"—maybe "cut the 12-foot piece to"—the options are "11 feet", "1 foot", "2 feet". Wait, let's re-express:

Original sides: \(5\), \(6\), \(12\). Triangle inequality fails because \(5 + 6 < 12\). So we need to make the longest side (12) shorter so that \(5 + 6 > \text{new longest side}\). Let the new length of the 12-foot piece be \(x\). Then \(5 + 6 > x \implies x < 11\). Also, \(x\) must be greater than \(6 - 5 = 1\) (from \(x + 5 > 6\)). So \(1 < x < 11\).

Now, the options for \(x\) are \(11\), \(1\), \(2\). Let's check:

- \(x = 11\): \(5 + 6 = 11\), which is not greater than \(11\) (equal), so invalid.
- \(x = 1\): \(5 + 1 = 6\), not greater than \(6\) (equal), invalid.
- \(x = 2\): \(5 + 2 = 7 > 6\), \(6 + 2 = 8 > 5\), \(5 + 6 = 11 > 2\). All inequalities hold. So \(x = 2\) is valid.

Wait, but the problem says "cut the 12-foot piece by"—maybe "cut off" a certain length. If we cut off \(10\) feet from the \(12\)-foot piece, we get \(2\) feet. But the options are "11 feet", "1 foot", "2 feet" as the new length? Wait, the image shows the drop-down with "11 feet", "1 foot", "2 feet" (the visible text is "11 feet", "1 foot", "2 feet" in the drop-down). So the correct new length is \(2\) feet? Wait, no—wait, maybe the first part is "cannot" (since the original sides don't work), and the second part is cutting the \(12\)-foot piece to \(2\) feet? Wait, but let's confirm:

If we cut the \(12\)-foot piece to \(2\) feet, the sides are \(5\), \(6\), \(2\). Let's check all three inequalities:

1. \(5 + 6 > 2\): \(11 > 2\) (true)
2. \(5 + 2 > 6\): \(7 > 6\) (true)
3. \(6 + 2 > 5\): \(8 > 5\) (true)

Yes, that works. If we cut it to \(11\) feet, the sides are \(5\), \(6\), \(11\):

1. \(5 + 6 > 11\): \(11 > 11\) (false, it's equal)
2. \(5 + 11 > 6\): \(16 > 6\) (true)
3. \(6 + 11 > 5\): \(17 > 5\) (true)

But the first inequality fails, so that's invalid. If we cut it to \(1\) foot, the sides are \(5\), \(6\), \(1\):

1. \(5 + 6 > 1\): \(11 > 1\) (true)
2. \(5 + 1 > 6\): \(6 > 6\) (false, equal)
3. \(6 + 1 > 5\): \(7 > 5\) (true)

Second inequality fails, so invalid. Therefore, the correct length to cut the \(12\)-foot piece to is \(2\) feet. Wait, but the options are "11 feet", "1 foot", "2 feet" as the new length? Wait, the image shows the drop-down with those options. So the correct answers are:

- First drop-down: "cannot"
- Second drop-down: "2 feet" (but wait, the options are "11 feet", "1 foot", "2 feet"—wait, maybe I misread. Wait, the text says "She could cut the 12-foot piece by"—maybe "cut the 12-foot piece to"—the options are "11 feet", "1 foot", "2 feet". So the correct new length is \(2\) feet? Wait, no—wait, the problem says "cut the 12-foot piece by"—maybe "cut off" a length. For example, cut off \(10\) feet (leaving \(2\) feet), cut off \(1\) foot (leaving \(11\) feet), or cut off \(11\) feet (leaving \(1\) foot). So the "by" refers to how much to cut off. So:

- Cut off \(10\) feet: leave \(2\) feet (valid)
- Cut off \(1\) foot: leave \(11\) feet (invalid)
- Cut off \(11\) feet: leave \(1\) foot (invalid)

But the options are "11 feet", "1 foot", "2 feet"—maybe the options are the length to cut off, not the remaining length. Wait, this is confusing. But based on the triangle inequality, the only way to make the triangle is to have the longest side be less than \(11\) (since \(5 + 6 = 11\)). So the longest side must be less than \(11\), so we need to cut the \(12\)-foot piece to a length less than \(11\). The only option less than \(11\) (among \(11\), \(1\), \(2\)) is \(1\) and \(2\). But \(1\) is too short (since \(5 + 1 = 6\), not greater). So \(2\) is the correct length. Therefore, the carpenter cannot build the frame with the original pieces, and should cut the \(12\)-foot piece to \(2\) feet.

Answer:

First drop-down: cannot
Second drop-down: 2 feet (from the options: the option with "2 feet")