Answered

Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

What are the coordinates of point [tex]\( P \)[/tex] on the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that [tex]\( P \)[/tex] is [tex]\(\frac{1}{4}\)[/tex] the length of the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]?

A. [tex]\(\left(\frac{-29}{4}, \frac{-3}{2}\right)\)[/tex]
B. [tex]\(\left(\frac{-13}{4}, \frac{1}{2}\right)\)[/tex]
C. [tex]\(\left(\frac{-11}{4}, \frac{-1}{2}\right)\)[/tex]
D. [tex]\(\left(\frac{25}{4}, \frac{-1}{2}\right)\)[/tex]

Sagot :

GinnyAnswer
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]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.