Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
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].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
