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Answer 1

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ANALYTICAL GEOMETRY -

A rectangle bisects the given isosceles triangle , respectively at heights m ( PQ ) and m ( SR ) on AB and AC

Bisected side lengths -
AB = 2AP = 2PB
AP = PB = 10 UNITS
Similarly on the other similar side of isosceles triangle ,
AC = 2AS = 2SC
AS = SC = 10 UNITS

REMEMBER THAT BISECTION OF SIDE LENGTHS OF ISOSCELES TRIANGLE IS DEPENDENT ON THE SIDE LENGTHS OF RECTANGULAR FIGURE

So ,
BC = BQ + QR + RC = 24 UNITS
QR = 2n
BQ = RC

PERPENDICULAR HEIGHT FROM ANGLE A IS BISECTING THE SIDE LENGTH BC ,
HENCE ,

BISECTED SIDE LENGTHS -
BQ = QD = DR = RC = 6 UNITS

NOW IN RIGHT ANGLE TRIANGLE PQB ,
PQ² + BQ² = PB² [ PQ = m = SR ]
PQ² = PB² - BQ² [ PB = 10 , BQ = 6 UNITS ]
PQ² = 100 - 36 = 64 UNITS
PQ = 8 UNITS = m

HENCE , RESOLVED SIDE LENGTHS ARE -
m = 8 UNITS
n = 6 UNITS

NOW VERIFY THE RESULTS AT THE EXPRESSION PROVIDED

$m = 16 - \frac{4n}{3} \\ for \: m = 8 \: and \: n = 6 \\ 8 = 16 - 4( \frac{6}{3}) \\ \\ 8 = 16 - 4(2) \\ lhs = rhs \\ hence \: proved$
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