Question #316374

A line segment has end points A (-12, -11) and B (6, 19). Find the three points that divide AB into four equal pieces. A) show that the line segment with end points A (5,3) and B (5 ,-5) is a chord of the circle X to the power of 2 + Y to the power of two = 34


1
Expert's answer
2022-03-23T16:59:20-0400

To determine the three points that divide AB into 4 equal parts, we shall find a point X on AB such that X is the midpoint of AB. Then we shall find a point Y on AX such that Y is the midpoint of AX. Finally, we shall find a point Z such that Z is the midpoint of XB


Midpoint of AB == (x,y)(x,y) == (12+62,11+192)(\frac{-12+6}{2}, \frac{-11+19}{2})

Midpoint of AB =(x,y)=(3,4)= (x,y) = (-3, 4)


X(3,4)X(-3,4)


Midpoint of AX =(x,y)=(1232,11+42)= (x,y) = (\frac{-12-3}{2}, \frac{-11+4}{2})

Midpoint of AX =(x,y)=(152,72)= (x,y) = (\frac{-15}{2}, \frac{-7}{2})


Y(152,72)Y(\frac{-15}{2}, \frac{-7}{2})



Midpoint of BX =(x,y)=(3+62,4+192)= (x,y) = (\frac{-3+6}{2}, \frac{4+19}{2})

Midpoint of BX =(x,y)=(32,232)= (x,y) = (\frac{3}{2}, \frac{23}{2})


Z(32,232)Z(\frac{3}{2}, \frac{23}{2})




Equation of circle : x2+y2=34x²+y²=34


To show that line AB is a chord of the circle, we shall show that one of the points lie on the circle, and the midpoint of line AB is not the centre of the circle


Let x=5,y=3x=-5, y=3

x2+y2=(5)2+32=25+9=34x²+y²=(-5)²+3²=25+9=34


Hence, A(5,3)A(-5,3) lies on the circle


Midpoint of AB =(x,y)=(5+52,5+32)= (x,y) = (\frac{5+5}{2}, \frac{-5+3}{2})

Midpoint of AB =(x,y)=(5,1)= (x,y) = (5, -1)


From x2+y2=34x²+y²=34 , it is evident that the centre of the circle is (0,0)(0,0)


(0,0)(5,1)(0,0) ≠ (5, -1)


Hence, line AB is a chord of the circle, x2+y2=34x²+y²=34



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