Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Certainly! Let's solve the problem step-by-step.
We are given three points:
- [tex]\( E(-1, d) \)[/tex]
- [tex]\( F(1, 1) \)[/tex]
- [tex]\( G(3, -5) \)[/tex]
It is also given that point [tex]\( F \)[/tex] is the midpoint of the line segment [tex]\( EG \)[/tex].
### Step 1: Using the Midpoint Formula
The coordinates of the midpoint [tex]\( M \)[/tex] of a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Here, point [tex]\( F \)[/tex] is the midpoint of [tex]\( E \)[/tex] and [tex]\( G \)[/tex]. Therefore:
[tex]\[ F = \left( \frac{-1 + 3}{2}, \frac{d + (-5)}{2} \right) \][/tex]
Given that [tex]\( F(1, 1) \)[/tex], we can now set up the equations as follows:
### Step 2: Set up the equations for the x-coordinates
[tex]\[ 1 = \frac{-1 + 3}{2} \][/tex]
### Step 3: Solve for the x-coordinates to verify
[tex]\[ 1 = \frac{2}{2} \][/tex]
[tex]\[ 1 = 1 \][/tex]
The x-coordinate part is verified indeed. Now, let's move on to the y-coordinates.
### Step 4: Set up the equations for the y-coordinates
[tex]\[ 1 = \frac{d + (-5)}{2} \][/tex]
### Step 5: Solve for [tex]\( d \)[/tex]
[tex]\[ 1 = \frac{d - 5}{2} \][/tex]
[tex]\[ 2 \times 1 = d - 5 \][/tex]
[tex]\[ 2 = d - 5 \][/tex]
Add 5 to both sides of the equation:
[tex]\[ 2 + 5 = d \][/tex]
[tex]\[ d = 7 \][/tex]
### Conclusion
The value of [tex]\( d \)[/tex] is [tex]\( \boxed{7} \)[/tex].
We are given three points:
- [tex]\( E(-1, d) \)[/tex]
- [tex]\( F(1, 1) \)[/tex]
- [tex]\( G(3, -5) \)[/tex]
It is also given that point [tex]\( F \)[/tex] is the midpoint of the line segment [tex]\( EG \)[/tex].
### Step 1: Using the Midpoint Formula
The coordinates of the midpoint [tex]\( M \)[/tex] of a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Here, point [tex]\( F \)[/tex] is the midpoint of [tex]\( E \)[/tex] and [tex]\( G \)[/tex]. Therefore:
[tex]\[ F = \left( \frac{-1 + 3}{2}, \frac{d + (-5)}{2} \right) \][/tex]
Given that [tex]\( F(1, 1) \)[/tex], we can now set up the equations as follows:
### Step 2: Set up the equations for the x-coordinates
[tex]\[ 1 = \frac{-1 + 3}{2} \][/tex]
### Step 3: Solve for the x-coordinates to verify
[tex]\[ 1 = \frac{2}{2} \][/tex]
[tex]\[ 1 = 1 \][/tex]
The x-coordinate part is verified indeed. Now, let's move on to the y-coordinates.
### Step 4: Set up the equations for the y-coordinates
[tex]\[ 1 = \frac{d + (-5)}{2} \][/tex]
### Step 5: Solve for [tex]\( d \)[/tex]
[tex]\[ 1 = \frac{d - 5}{2} \][/tex]
[tex]\[ 2 \times 1 = d - 5 \][/tex]
[tex]\[ 2 = d - 5 \][/tex]
Add 5 to both sides of the equation:
[tex]\[ 2 + 5 = d \][/tex]
[tex]\[ d = 7 \][/tex]
### Conclusion
The value of [tex]\( d \)[/tex] is [tex]\( \boxed{7} \)[/tex].
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.