Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine the coordinates of point [tex]\( P \)[/tex] such that it lies [tex]\(\frac{1}{4}\)[/tex] of the way along the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], we need the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Let's use [tex]\( A \left(\frac{-29}{4}, \frac{-3}{2}\right) \)[/tex] and [tex]\( B \left(\frac{25}{4}, \frac{-1}{2}\right) \)[/tex].
### Step 1: Calculate the distance vector [tex]\(\overrightarrow{AB}\)[/tex]
First, we find the vector [tex]\(\overrightarrow{AB}\)[/tex]. This vector is obtained by subtracting the coordinates of [tex]\( A \)[/tex] from [tex]\( B \)[/tex]:
[tex]\[ \overrightarrow{AB} = \left( \frac{25}{4} - \frac{-29}{4}, \frac{-1}{2} - \frac{-3}{2} \right) \][/tex]
Simplify each component:
[tex]\[ \overrightarrow{AB}_x = \frac{25}{4} + \frac{29}{4} = \frac{54}{4} = 13.5 \][/tex]
[tex]\[ \overrightarrow{AB}_y = \frac{-1}{2} + \frac{3}{2} = 1.0 \][/tex]
So, the vector [tex]\(\overrightarrow{AB}\)[/tex] is:
[tex]\[ \overrightarrow{AB} = (13.5, 1.0) \][/tex]
### Step 2: Calculate the coordinates of point [tex]\( P \)[/tex]
Point [tex]\( P \)[/tex] is located [tex]\(\frac{1}{4}\)[/tex] of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]. To find the coordinates of [tex]\( P \)[/tex], we start at point [tex]\( A \)[/tex] and move [tex]\(\frac{1}{4}\)[/tex] of the way along the vector [tex]\(\overrightarrow{AB}\)[/tex]:
[tex]\[ P_x = A_x + \frac{1}{4} \cdot \overrightarrow{AB}_x \][/tex]
[tex]\[ P_y = A_y + \frac{1}{4} \cdot \overrightarrow{AB}_y \][/tex]
Using the values we have:
[tex]\[ P_x = \frac{-29}{4} + \frac{1}{4} \cdot 13.5 \][/tex]
[tex]\[ P_y = \frac{-3}{2} + \frac{1}{4} \cdot 1.0 \][/tex]
Simplify these expressions:
[tex]\[ P_x = \frac{-29}{4} + \frac{13.5}{4} = \frac{-29 + 13.5}{4} = \frac{-15.5}{4} = -3.875 \][/tex]
[tex]\[ P_y = \frac{-3}{2} + \frac{1}{4} \cdot 1 = \frac{-3}{2} + \frac{1}{4} = \frac{-6}{4} + \frac{1}{4} = \frac{-5}{4} = -1.25 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ P(-3.875, -1.25) \][/tex]
### Step 1: Calculate the distance vector [tex]\(\overrightarrow{AB}\)[/tex]
First, we find the vector [tex]\(\overrightarrow{AB}\)[/tex]. This vector is obtained by subtracting the coordinates of [tex]\( A \)[/tex] from [tex]\( B \)[/tex]:
[tex]\[ \overrightarrow{AB} = \left( \frac{25}{4} - \frac{-29}{4}, \frac{-1}{2} - \frac{-3}{2} \right) \][/tex]
Simplify each component:
[tex]\[ \overrightarrow{AB}_x = \frac{25}{4} + \frac{29}{4} = \frac{54}{4} = 13.5 \][/tex]
[tex]\[ \overrightarrow{AB}_y = \frac{-1}{2} + \frac{3}{2} = 1.0 \][/tex]
So, the vector [tex]\(\overrightarrow{AB}\)[/tex] is:
[tex]\[ \overrightarrow{AB} = (13.5, 1.0) \][/tex]
### Step 2: Calculate the coordinates of point [tex]\( P \)[/tex]
Point [tex]\( P \)[/tex] is located [tex]\(\frac{1}{4}\)[/tex] of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]. To find the coordinates of [tex]\( P \)[/tex], we start at point [tex]\( A \)[/tex] and move [tex]\(\frac{1}{4}\)[/tex] of the way along the vector [tex]\(\overrightarrow{AB}\)[/tex]:
[tex]\[ P_x = A_x + \frac{1}{4} \cdot \overrightarrow{AB}_x \][/tex]
[tex]\[ P_y = A_y + \frac{1}{4} \cdot \overrightarrow{AB}_y \][/tex]
Using the values we have:
[tex]\[ P_x = \frac{-29}{4} + \frac{1}{4} \cdot 13.5 \][/tex]
[tex]\[ P_y = \frac{-3}{2} + \frac{1}{4} \cdot 1.0 \][/tex]
Simplify these expressions:
[tex]\[ P_x = \frac{-29}{4} + \frac{13.5}{4} = \frac{-29 + 13.5}{4} = \frac{-15.5}{4} = -3.875 \][/tex]
[tex]\[ P_y = \frac{-3}{2} + \frac{1}{4} \cdot 1 = \frac{-3}{2} + \frac{1}{4} = \frac{-6}{4} + \frac{1}{4} = \frac{-5}{4} = -1.25 \][/tex]
Therefore, the coordinates of point [tex]\( P \)[/tex] are:
[tex]\[ P(-3.875, -1.25) \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.