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Grace has a segment with endpoints [tex]\( C(3,4) \)[/tex] and [tex]\( D(11,3) \)[/tex] that is divided by a point [tex]\( E \)[/tex] in a [tex]\( 3:5 \)[/tex] ratio. The distance between the [tex]\( x \)[/tex]-coordinates is 8 units.

Which fraction will help Grace find the [tex]\( x \)[/tex]-coordinate for point [tex]\( E \)[/tex]?

A. [tex]\(\frac{3}{5}\)[/tex]

B. [tex]\(\frac{5}{3}\)[/tex]

C. [tex]\(\frac{3}{8}\)[/tex]

D. [tex]\(\frac{5}{8}\)[/tex]

Sagot :

GinnyAnswer
To solve this problem, we need to determine the correct fraction of the total horizontal distance between points [tex]\( C \)[/tex] and [tex]\( D \)[/tex] that corresponds to the portion [tex]\( CE \)[/tex].

Given:
- The coordinates of point [tex]\( C \)[/tex] are [tex]\( (3, 4) \)[/tex].
- The coordinates of point [tex]\( D \)[/tex] are [tex]\( (11, 3) \)[/tex].
- The ratio [tex]\( CE : DE \)[/tex] is [tex]\( 3 : 5 \)[/tex].

First, let's find the total ratio. The total ratio is obtained by adding the individual parts of the ratio:
[tex]\[ 3 + 5 = 8 \][/tex]

Next, we determine the fraction of the total horizontal distance between [tex]\( C \)[/tex] and [tex]\( D \)[/tex] that corresponds to [tex]\( CE \)[/tex]. Since [tex]\( CE \)[/tex] is the portion that we are interested in, its fraction is calculated as follows:
[tex]\[ \text{Fraction of } CE = \frac{3}{8} \][/tex]

Therefore, the fraction that Grace should use to find the [tex]\( x \)[/tex]-coordinate of point [tex]\( E \)[/tex] is:
[tex]\[ \frac{3}{8} \][/tex]

Thus, the correct answer is:
[tex]\[ \frac{3}{8} \][/tex]
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