Math

QuestionFind point BB on line segment AC\overline{AC} where ABAB is 5 times BCBC. A=(7,5)A=(-7,5), C=(5,1)C=(5,-1).

Studdy Solution

STEP 1

Assumptions1. Point A is at (-7,5) . Point C is at (5,-1)
3. AB is5 times as long as BC. This means that B divides AC in the ratio51.

STEP 2

We can use the section formula to find the coordinates of point B. The section formula is given byx=m1x2+m2x1m1+m2x = \frac{m1x2 + m2x1}{m1 + m2}y=m1y2+m2y1m1+m2y = \frac{m1y2 + m2y1}{m1 + m2}where (x1,y1)(x1, y1) and (x2,y2)(x2, y2) are the coordinates of the two points, and m1m1 and m2m2 are the ratios in which the line segment is divided.

STEP 3

In this case, point A is (7,5)(-7,5) and point C is (5,1)(5,-1). The ratio in which B divides AC is51. So, m1=5m1 =5 and m2=1m2 =1.

STEP 4

Substitute the given values into the section formula to find the x-coordinate of point B.
x=+17+1x = \frac{ \cdot +1 \cdot -7}{ +1}

STEP 5

Calculate the x-coordinate of point B.
x=257=3x = \frac{25 -7}{} =3

STEP 6

Substitute the given values into the section formula to find the y-coordinate of point B.
y=51+155+1y = \frac{5 \cdot -1 +1 \cdot5}{5 +1}

STEP 7

Calculate the y-coordinate of point B.
y=5+56=0y = \frac{-5 +5}{6} =0The coordinates of point B are (3,0)(3,0).

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