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Sagot :
To find the coordinates of point [tex]\( Q \)[/tex] given the points [tex]\( P \)[/tex] and [tex]\( R \)[/tex], and the ratio [tex]\( PR : RQ = 2 : 3 \)[/tex], we can follow these steps:
### Step 1: Understanding the Given Information
1. Coordinates of [tex]\( P \)[/tex]: [tex]\((-10, 3)\)[/tex]
2. Coordinates of [tex]\( R \)[/tex]: [tex]\((4, 7)\)[/tex]
3. Ratio [tex]\( PR : RQ = 2 : 3 \)[/tex]. This means that:
- [tex]\( PR = 2 \)[/tex] parts
- [tex]\( RQ = 3 \)[/tex] parts
### Step 2: Set Up the Section Formula
Since [tex]\( R \)[/tex] divides [tex]\( PQ \)[/tex] in the ratio 2:3, the section formula for coordinates of [tex]\( R \)[/tex] is:
[tex]\[ R = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \][/tex]
However, we already know the coordinates of [tex]\( R \)[/tex]. Now, we need to determine the coordinates of [tex]\( Q \)[/tex] using the relation from [tex]\( R \)[/tex] and the given ratios.
### Step 3: Use the Ratios to Find Coordinates of [tex]\( Q \)[/tex]
We can determine the coordinates of [tex]\( Q \)[/tex] starting from the known point [tex]\( R = (4, 7) \)[/tex] and using the ratio backward to [tex]\( P \)[/tex].
### Step 4: Calculate [tex]\( x \)[/tex] Coordinate of [tex]\( Q \)[/tex]
We use the following backward ratio formula deriving from [tex]\( R \)[/tex] to find [tex]\( Q \)[/tex]:
[tex]\[ x_Q = \frac{x_R \cdot n - x_P \cdot m}{n} \][/tex]
Substituting values:
[tex]\[ x_Q = \frac{4 \cdot 3 - (-10) \cdot 2}{3} \][/tex]
[tex]\[ x_Q = \frac{12 + 20}{3} \][/tex]
[tex]\[ x_Q = \frac{32}{3} \][/tex]
[tex]\[ x_Q \approx 10.67 \][/tex]
### Step 5: Calculate [tex]\( y \)[/tex] Coordinate of [tex]\( Q \)[/tex]
Similarly, for [tex]\( y \)[/tex] coordinate:
[tex]\[ y_Q = \frac{y_R \cdot n - y_P \cdot m}{n} \][/tex]
Substituting values:
[tex]\[ y_Q = \frac{7 \cdot 3 - 3 \cdot 2}{3} \][/tex]
[tex]\[ y_Q = \frac{21 - 6}{3} \][/tex]
[tex]\[ y_Q = \frac{15}{3} \][/tex]
[tex]\[ y_Q = 5 \][/tex]
### Step 6: Combine the Coordinates
Now, [tex]\( Q \)[/tex] has the coordinates:
[tex]\[ Q = (10.67, 5.0) \][/tex]
Therefore, the coordinates of point [tex]\( Q \)[/tex] are [tex]\((10.67, 5.0)\)[/tex]. The closest answer option to these coordinates is not provided in the choices given in the problem statement. Based on our detailed calculations, we confirm the accurate coordinates of [tex]\( Q \)[/tex] to be approximately [tex]\((10.67, 5.0)\)[/tex].
### Step 1: Understanding the Given Information
1. Coordinates of [tex]\( P \)[/tex]: [tex]\((-10, 3)\)[/tex]
2. Coordinates of [tex]\( R \)[/tex]: [tex]\((4, 7)\)[/tex]
3. Ratio [tex]\( PR : RQ = 2 : 3 \)[/tex]. This means that:
- [tex]\( PR = 2 \)[/tex] parts
- [tex]\( RQ = 3 \)[/tex] parts
### Step 2: Set Up the Section Formula
Since [tex]\( R \)[/tex] divides [tex]\( PQ \)[/tex] in the ratio 2:3, the section formula for coordinates of [tex]\( R \)[/tex] is:
[tex]\[ R = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \][/tex]
However, we already know the coordinates of [tex]\( R \)[/tex]. Now, we need to determine the coordinates of [tex]\( Q \)[/tex] using the relation from [tex]\( R \)[/tex] and the given ratios.
### Step 3: Use the Ratios to Find Coordinates of [tex]\( Q \)[/tex]
We can determine the coordinates of [tex]\( Q \)[/tex] starting from the known point [tex]\( R = (4, 7) \)[/tex] and using the ratio backward to [tex]\( P \)[/tex].
### Step 4: Calculate [tex]\( x \)[/tex] Coordinate of [tex]\( Q \)[/tex]
We use the following backward ratio formula deriving from [tex]\( R \)[/tex] to find [tex]\( Q \)[/tex]:
[tex]\[ x_Q = \frac{x_R \cdot n - x_P \cdot m}{n} \][/tex]
Substituting values:
[tex]\[ x_Q = \frac{4 \cdot 3 - (-10) \cdot 2}{3} \][/tex]
[tex]\[ x_Q = \frac{12 + 20}{3} \][/tex]
[tex]\[ x_Q = \frac{32}{3} \][/tex]
[tex]\[ x_Q \approx 10.67 \][/tex]
### Step 5: Calculate [tex]\( y \)[/tex] Coordinate of [tex]\( Q \)[/tex]
Similarly, for [tex]\( y \)[/tex] coordinate:
[tex]\[ y_Q = \frac{y_R \cdot n - y_P \cdot m}{n} \][/tex]
Substituting values:
[tex]\[ y_Q = \frac{7 \cdot 3 - 3 \cdot 2}{3} \][/tex]
[tex]\[ y_Q = \frac{21 - 6}{3} \][/tex]
[tex]\[ y_Q = \frac{15}{3} \][/tex]
[tex]\[ y_Q = 5 \][/tex]
### Step 6: Combine the Coordinates
Now, [tex]\( Q \)[/tex] has the coordinates:
[tex]\[ Q = (10.67, 5.0) \][/tex]
Therefore, the coordinates of point [tex]\( Q \)[/tex] are [tex]\((10.67, 5.0)\)[/tex]. The closest answer option to these coordinates is not provided in the choices given in the problem statement. Based on our detailed calculations, we confirm the accurate coordinates of [tex]\( Q \)[/tex] to be approximately [tex]\((10.67, 5.0)\)[/tex].
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