Question
△LMN where LM is a vertical line segment whose length is a multiple of 3 and MN is a horizontal line segment whose length is a multiple of 6 plot point q on LM such that LQ / Qm = 1 / 2. Similarly , plot point R on MN such that NR / RM = 1 / 2. Connect points Q and R to create QR. detailed
Answers
GPT-4o mini
To create the triangle \( \triangle LMN \) and plot points \( Q \) and \( R \), we will follow these steps:
1. **Draw Line Segment LM**:
- Let \( L \) be at the coordinates \( (0, 0) \) and \( M \) at \( (0, h) \), where \( h \) is a multiple of 3. For example, let \( h = 6 \), so \( M \) is at \( (0, 6) \).
2. **Draw Line Segment MN**:
- Since \( MN \) is horizontal, let \( N \) be at \( (b, 6) \), where \( b \) is a multiple of 6. Let’s say \( b = 12 \), so \( N \) is at \( (12, 6) \).
3. **Plot Point Q on LM**:
- Since \( Q \) is located on \( LM \) such that the ratio \( \frac{LQ}{QM} = \frac{1}{2} \), let's denote the length \( LQ = x \) and \( QM = 2x \).
- The total length \( LM \) is \( LQ + QM = x + 2x = 3x \). Since \( LM = 6\) (which is a multiple of 3, fitting our requirements), we can set \( 3x = 6 \), so \( x = 2 \).
- Thus, the coordinates of \( Q \) are \( (0, 2) \).
4. **Plot Point R on MN**:
- Now for point \( R \) on \( MN \), we need the ratio \( \frac{NR}{RM} = \frac{1}{2} \). Let’s denote \( NR = y \) and \( RM = 2y \).
- The total length \( NM \) is \( NR + RM = y + 2y = 3y \). Since \( MN = 12\) (which is a multiple of 6), we can set \( 3y = 12 \), so \( y = 4 \).
- Thus, the coordinates of \( R \) are \( (12 - 4, 6) = (8, 6) \).
5. **Connect Points Q and R**:
- You will now connect points \( Q(0, 2) \) and \( R(8, 6) \) with a straight line to form segment \( QR \).
### Summary of Coordinates:
- \( L(0, 0) \)
- \( M(0, 6) \)
- \( N(12, 6) \)
- \( Q(0, 2) \)
- \( R(8, 6) \)
### Visual Representation:
```
N(12, 6) −−−−−−−−−−−−−
| |
| |
Q(0, 2) |
| |
| |
L(0, 0) −−−−−−−−−−−−−
M(0, 6)
```
With these coordinates, you have correctly plotted the triangle \( \triangle LMN \) and established the points \( Q \) and \( R \) to connect and form line segment \( QR \).
1. **Draw Line Segment LM**:
- Let \( L \) be at the coordinates \( (0, 0) \) and \( M \) at \( (0, h) \), where \( h \) is a multiple of 3. For example, let \( h = 6 \), so \( M \) is at \( (0, 6) \).
2. **Draw Line Segment MN**:
- Since \( MN \) is horizontal, let \( N \) be at \( (b, 6) \), where \( b \) is a multiple of 6. Let’s say \( b = 12 \), so \( N \) is at \( (12, 6) \).
3. **Plot Point Q on LM**:
- Since \( Q \) is located on \( LM \) such that the ratio \( \frac{LQ}{QM} = \frac{1}{2} \), let's denote the length \( LQ = x \) and \( QM = 2x \).
- The total length \( LM \) is \( LQ + QM = x + 2x = 3x \). Since \( LM = 6\) (which is a multiple of 3, fitting our requirements), we can set \( 3x = 6 \), so \( x = 2 \).
- Thus, the coordinates of \( Q \) are \( (0, 2) \).
4. **Plot Point R on MN**:
- Now for point \( R \) on \( MN \), we need the ratio \( \frac{NR}{RM} = \frac{1}{2} \). Let’s denote \( NR = y \) and \( RM = 2y \).
- The total length \( NM \) is \( NR + RM = y + 2y = 3y \). Since \( MN = 12\) (which is a multiple of 6), we can set \( 3y = 12 \), so \( y = 4 \).
- Thus, the coordinates of \( R \) are \( (12 - 4, 6) = (8, 6) \).
5. **Connect Points Q and R**:
- You will now connect points \( Q(0, 2) \) and \( R(8, 6) \) with a straight line to form segment \( QR \).
### Summary of Coordinates:
- \( L(0, 0) \)
- \( M(0, 6) \)
- \( N(12, 6) \)
- \( Q(0, 2) \)
- \( R(8, 6) \)
### Visual Representation:
```
N(12, 6) −−−−−−−−−−−−−
| |
| |
Q(0, 2) |
| |
| |
L(0, 0) −−−−−−−−−−−−−
M(0, 6)
```
With these coordinates, you have correctly plotted the triangle \( \triangle LMN \) and established the points \( Q \) and \( R \) to connect and form line segment \( QR \).