Math / Geometry

QuestionThe coordinates of the endpoints of $\overline{S T}$ are $S(10,2)$ and $T(17,16)$ . Point $U$ is on $\overline{S T}$ and divides it such that $S U: T U$ is $2: 5$ .
What are the coordinates of $U$ ? Write your answers as integers or decimals. $\square$ $\square$ Submit

Studdy Solution

STEP 1

1. The coordinates of point $S$ are $(10, 2)$ .
2. The coordinates of point $T$ are $(17, 16)$ .
3. Point $U$ divides the segment $\overline{ST}$ in the ratio $2:5$ .

STEP 2

1. Recall the section formula for internal division of a line segment.
2. Substitute the given values into the section formula.
3. Calculate the coordinates of point $U$ .

STEP 3

Recall the section formula for internal division of a line segment. If a point $U(x, y)$ divides the line segment joining $S(x_1, y_1)$ and $T(x_2, y_2)$ in the ratio $m:n$ , then the coordinates of $U$ are given by:
\[ x = \frac{mx_2 + nx_1}{m+n} $ \[ y = \frac{my_2 + ny_1}{m+n} $

STEP 4

Substitute the given values into the section formula. Here, $m = 2$ , $n = 5$ , $x_1 = 10$ , $y_1 = 2$ , $x_2 = 17$ , $y_2 = 16$ .
For $x$ : \[ x = \frac{2 \times 17 + 5 \times 10}{2+5} $
For $y$ : \[ y = \frac{2 \times 16 + 5 \times 2}{2+5} $

STEP 5

Calculate the coordinates of point $U$ .
For $x$ : \[ x = \frac{34 + 50}{7} = \frac{84}{7} = 12 $
For $y$ : \[ y = \frac{32 + 10}{7} = \frac{42}{7} = 6 $
The coordinates of point $U$ are $(12, 6)$ .
The coordinates of $U$ are:
$\boxed{12}$ $\boxed{6}$

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