Question

Determine the location of point L
if it divides AB¯¯¯¯¯¯¯¯
into two segments in the ratio 1:3.

(1 point)
Responses

(14,14)
left parenthesis Start Fraction 1 over 4 End Fraction comma Start Fraction 1 over 4 End Fraction right parenthesis

(−23,1)
left parenthesis negative Start Fraction 2 over 3 End Fraction comma 1 right parenthesis

(1,−32)
left parenthesis 1 comma negative Start Fraction 3 over 2 End Fraction right parenthesis

(−12,34)
A(0,0) B(-2,3)

Answer 1

To find the coordinates of point L that divides the segment AB in the ratio 1:3, we can use the section formula.

Let A = (0, 0) and B = (-2, 3). The ratio \( m:n = 1:3 \).

The formula for finding the coordinates of point L that divides the segment AB in the ratio m:n is given by:

\[
L\left(x, y\right) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Here:
- \( (x_1, y_1) \) are the coordinates of point A (0, 0)
- \( (x_2, y_2) \) are the coordinates of point B (-2, 3)
- \( m = 1 \) and \( n = 3 \)

Now, substituting the values into the formula, we have:

\[
L\left(x, y\right) = \left( \frac{1 \cdot -2 + 3 \cdot 0}{1 + 3}, \frac{1 \cdot 3 + 3 \cdot 0}{1 + 3} \right)
\]

Calculating the x-coordinate:

\[
x = \frac{-2 + 0}{4} = \frac{-2}{4} = -\frac{1}{2}
\]

Calculating the y-coordinate:

\[
y = \frac{3 + 0}{4} = \frac{3}{4}
\]

Thus, the coordinates of point L are:

\[
L\left(-\frac{1}{2}, \frac{3}{4}\right)
\]

Now let's check the options provided in the question. None of them correspond to \( \left(-\frac{1}{2}, \frac{3}{4}\right) \). So there may have been a misunderstanding in the ratio definition from your question or a misalignment with the provided options.

However, if the ratio is indeed \( 1:3 \) and points A and B are as stated, \( L\) is located at these coordinates. To ensure a correct result, you may want to recheck the options provided.

If you have further details or options to verify against, please provide them!