Triangle abc has vertices a(1,5) b(9,7) and c(7,13) m is the midpoint of ab and n is the midpoint of bc. which of the following is the length of mn? a.3 b.4 c.5 d.6

By Mid Point Formula ,

In the cordinates of Mid point ,



Applying formula for pt. M

Cordinates of A = ( 1 , 5 )
Cordinates of B = ( 9 , 7 )

So,





So, M = ( 5 , 6 )

Now applying formula for pt. N

Cordinates of B = ( 9 , 7 )
Cordinates of C = ( 7 , 13 )

So,





N = ( 8 , 10 )

Now ,

By Distance formula ,

M = ( 5 , 6 )



N = ( 8 , 10 )



Therefore,






Distance = √25

Distance = 5 units

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12