Question

triangle - side relation dm
due oct 31 by 11:59pm points 100
submitting an external tool
score: 6/10 penalty: 1 off
angle side relationship
question
in \\( \delta \mathrm{tuv}, \mathrm{uv}=12, \mathrm{vt}=13 \\), and \\( \mathrm{tu}=4 \\). which
statement about the angles of \\( \delta \mathrm{tuv} \\) must be true?
answer
\\( \mathrm{m} \angle v>\mathrm{m} \angle u>\mathrm{m} \angle t \\)
\\( \mathrm{m} \angle t>\mathrm{m} \angle v>\mathrm{m} \angle u \\)
\\( \mathrm{m} \angle u>\mathrm{m} \angle t>\mathrm{m} \angle v \\)
\\( \mathrm{m} \angle t>\mathrm{m} \angle u>\mathrm{m} \angle v \\)
\\( \mathrm{m} \angle u>\mathrm{m} \angle v>\mathrm{m} \angle t \\)
\\( \mathrm{m} \angle v>\mathrm{m} \angle t>\mathrm{m} \angle u \\)

Explanation:

Step1: Recall the angle - side relationship in a triangle

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. That is, if in \(\triangle ABC\), \(a,b,c\) are the lengths of the sides opposite to \(\angle A,\angle B,\angle C\) respectively, then if \(a > b>c\), then \(m\angle A>m\angle B > m\angle C\).

Step2: Identify the sides of \(\triangle TUV\) and their opposite angles

In \(\triangle TUV\):
- The length of \(UV = 12\), and the angle opposite to \(UV\) is \(\angle T\) (because in \(\triangle TUV\), side \(UV\) is opposite \(\angle T\)).
- The length of \(VT=13\), and the angle opposite to \(VT\) is \(\angle U\) (side \(VT\) is opposite \(\angle U\)).
- The length of \(TU = 4\), and the angle opposite to \(TU\) is \(\angle V\) (side \(TU\) is opposite \(\angle V\)).

Step3: Compare the lengths of the sides

We are given \(UV = 12\), \(VT = 13\), and \(TU=4\). Comparing the lengths of the sides: \(VT>UV>TU\), i.e., \(13 > 12>4\).

Step4: Compare the measures of the angles using the angle - side relationship

Since \(VT>UV>TU\), and the angles opposite to these sides are \(\angle U\), \(\angle T\) and \(\angle V\) respectively, we have:
The side \(VT\) (length \(13\)) is opposite \(\angle U\), side \(UV\) (length \(12\)) is opposite \(\angle T\), and side \(TU\) (length \(4\)) is opposite \(\angle V\).
So, \(m\angle U>m\angle T > m\angle V\)? Wait, no, wait. Wait, let's re - check the opposite angles:
- Side \(UV\): connects \(U\) and \(V\), so it is opposite \(\angle T\).
- Side \(VT\): connects \(V\) and \(T\), so it is opposite \(\angle U\).
- Side \(TU\): connects \(T\) and \(U\), so it is opposite \(\angle V\).

So, side lengths: \(VT = 13\) (opposite \(\angle U\)), \(UV=12\) (opposite \(\angle T\)), \(TU = 4\) (opposite \(\angle V\)).
Since \(13>12 > 4\), the angles opposite to these sides will have the same order. So \(m\angle U>m\angle T>m\angle V\)? No, wait, no. Wait, the larger the side, the larger the angle opposite. So:
- Side \(VT = 13\) (longest side) is opposite \(\angle U\), so \(m\angle U\) is the largest angle among the three.
- Side \(UV = 12\) (second - longest side) is opposite \(\angle T\), so \(m\angle T\) is the second - largest angle.
- Side \(TU=4\) (shortest side) is opposite \(\angle V\), so \(m\angle V\) is the smallest angle.

Wait, but let's re - list the sides and their opposite angles:
- Side opposite \(\angle T\): \(UV = 12\)
- Side opposite \(\angle U\): \(VT=13\)
- Side opposite \(\angle V\): \(TU = 4\)

Since \(VT(13)>UV(12)>TU(4)\), then the angles opposite them: \(m\angle U>m\angle T>m\angle V\)? No, that's not one of the options. Wait, maybe I made a mistake in identifying the opposite angles.

Wait, let's label the triangle properly. Let's denote the vertices: \(T\), \(U\), \(V\). So the sides:
- \(TU\): between \(T\) and \(U\)
- \(UV\): between \(U\) and \(V\)
- \(VT\): between \(V\) and \(T\)

So, angle at \(T\) (\(\angle T\)) is between sides \(VT\) and \(TU\), so the side opposite \(\angle T\) is \(UV\).
Angle at \(U\) (\(\angle U\)) is between sides \(TU\) and \(UV\), so the side opposite \(\angle U\) is \(VT\).
Angle at \(V\) (\(\angle V\)) is between sides \(UV\) and \(VT\), so the side opposite \(\angle V\) is \(TU\).

So, side lengths: \(VT = 13\), \(UV = 12\), \(TU=4\). So \(VT>UV>TU\).

Therefore, the angles opposite to these sides:
- Opposite \(VT\) (length \(13\)) is \(\angle U\), so \(m\angle U\) is the largest.
- Opposite \(UV\) (length \(12\)) is \(\angle T\), so \(m\angle T\) is the second - largest.
- Opposite \(TU\) (length \(4\)) is \(\angle V\), so \(m\angle V\) is the smallest.

Wait, but looking at the options, we have an option \(m\angle T>m\angle U>m\angle V\)? No, wait the options are:
1. \(m\angle V>m\angle U>m\angle T\)
2. \(m\angle T>m\angle V>m\angle U\)
3. \(m\angle U>m\angle T>m\angle V\)
4. \(m\angle T>m\angle U>m\angle V\)
5. \(m\angle U>m\angle V>m\angle T\)
6. \(m\angle V>m\angle T>m\angle U\)

Wait, I think I made a mistake. Let's re - do the side - angle correspondence:

In \(\triangle TUV\):
- Side \(TU\): length \(4\), opposite \(\angle V\)
- Side \(UV\): length \(12\), opposite \(\angle T\)
- Side \(VT\): length \(13\), opposite \(\angle U\)

So the order of the sides from longest to shortest is \(VT(13)>UV(12)>TU(4)\)

So the angles opposite to these sides (from largest to smallest) should be the angles opposite to the longest to shortest sides. So:

Angle opposite \(VT\) (longest side) is \(\angle U\), so \(m\angle U\) is the largest.

Angle opposite \(UV\) (second - longest side) is \(\angle T\), so \(m\angle T\) is the second - largest.

Angle opposite \(TU\) (shortest side) is \(\angle V\), so \(m\angle V\) is the smallest.

So the order of the angles from largest to smallest is \(m\angle U>m\angle T>m\angle V\). But this option is not in the list? Wait, no, looking at the options again:

Wait the options are:

- \(m\angle V>m\angle U>m\angle T\)
- \(m\angle T>m\angle V>m\angle U\)
- \(m\angle U>m\angle T>m\angle V\)
- \(m\angle T>m\angle U>m\angle V\)
- \(m\angle U>m\angle V>m\angle T\)
- \(m\angle V>m\angle T>m\angle U\)

Wait, maybe I mixed up the opposite angles. Let's use the standard notation: in \(\triangle ABC\), side \(a\) is opposite \(\angle A\), side \(b\) opposite \(\angle B\), side \(c\) opposite \(\angle C\).

So in \(\triangle TUV\), let's denote:

- \(\angle T\) is at vertex \(T\), so the side opposite \(\angle T\) is \(UV\) (length \(12\))
- \(\angle U\) is at vertex \(U\), so the side opposite \(\angle U\) is \(VT\) (length \(13\))
- \(\angle V\) is at vertex \(V\), so the side opposite \(\angle V\) is \(TU\) (length \(4\))

So the lengths of the sides opposite the angles:

- Opposite \(\angle T\): \(12\)
- Opposite \(\angle U\): \(13\)
- Opposite \(\angle V\): \(4\)

So the order of the sides opposite the angles: \(13>12 > 4\) (opposite \(\angle U\), \(\angle T\), \(\angle V\) respectively)

So the order of the angles should be \(m\angle U>m\angle T>m\angle V\)

Wait, but looking at the options, the third option is \(m\angle U>m\angle T>m\angle V\). But maybe I made a mistake. Wait, let's check the option again. Wait, the user's options:

Wait the options are:

1. \(m\angle V>m\angle U>m\angle T\)
2. \(m\angle T>m\angle V>m\angle U\)
3. \(m\angle U>m\angle T>m\angle V\)
4. \(m\angle T>m\angle U>m\angle V\)
5. \(m\angle U>m\angle V>m\angle T\)
6. \(m\angle V>m\angle T>m\angle U\)

Wait, maybe I had the side - angle correspondence wrong. Let's think again. Let's list the sides:

\(TU = 4\), \(UV = 12\), \(VT=13\)

So the sides: \(VT\) is the longest (\(13\)), \(UV\) is next (\(12\)), \(TU\) is the shortest (\(4\))

Now, the angle opposite \(VT\) (longest side) is \(\angle U\) (because \(VT\) is between \(V\) and \(T\), so the angle at \(U\) is opposite \(VT\)).

The angle opposite \(UV\) (length \(12\)) is \(\angle T\) (because \(UV\) is between \(U\) and \(V\), so the angle at \(T\) is opposite \(UV\)).

The angle opposite \(TU\) (length \(4\)) is \(\angle V\) (because \(TU\) is between \(T\) and \(U\), so the angle at \(V\) is opposite \(TU\)).

So since \(VT>UV>TU\), then the angles opposite them: \(\angle U>\angle T>\angle V\) (because larger side opposite larger angle)

So \(m\angle U>m\angle T>m\angle V\), which is the third option.

Answer:

\(m\angle U>m\angle T>m\angle V\) (the third option in the list of answer choices)