Find the value of the variable and the length of each chord for #5 and #6.

  5.  y=6; DE=7; FG=8

  6.  x=4.2; JL=14.2; MN=13

Step-by-step explanation:

When chords cross inside a circle, the product of the segment lengths is a constant.

5. DH×EH = FH×GH

  y = GH = DH×EH/FH = 3×4/2 . . . . . divide by the coefficient of GH

  y = 6

Of course, the total chord length is the sum of the lengths of its segments.

  DE = DH +EH = 3+4 = 7

  FG = FH +GH = 2+6 = 8

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6. This problem works the same way as the previous one.

  x = LK = MK×NK/JK = 6×7/10 = 4.2

  JL = JK +LK = 10 +4.2 = 14.2

  MN = MK +NK = 6 +7 = 13

Comment on chords and secants

You can think of the points H (problem 5) and K (problem 6) as points where the chords meet. A chord is part of a secant line, a line that intersects the circle in two points. One can also consider these points (H or K) to be the points where the respective secant lines meet.

Then, the product that is a constant is the product of the distance from the meeting point (H or K) to one circle intersection with the distance from that meeting point to the other circle intersection. For problem 5, the constant is the product HD×HE = HF×HG.

It turns out that this rule regarding the products of lengths to points of intersection is also true when the secants intersect outside the circle.

That is, you only need to remember one rule to work both kinds of problems.

the answer would be d.

step-by-step explanation:

move all variables to the left side and all constants to the right side

(x−3)^2+(y−1)^2=16

this is the form of a circle. use this form to determine the center and radius of the circle.

( x − h ) 2 + ( y − k ) 2 = r 2

match the values in this circle to those of the standard form. the variable   r   represents the radius of the circle,   h   represents the x-offset from the origin, and   k   represents the y-offset from origin.

r = 4

h = 3

k = 1

the center of the circle is found at   ( h , k ) .

center:   ( 3 , 1 )

these values represent the important values for graphing and analyzing a circle.

center:   ( 3 , 1 ).

radius:   4