Given: klmn is a trapezoid m∠n = m∠kml me⊥kn , me = 3√5 , ke = 8, lm/kn = 3/5 find: km, lm, kn, area of klmn

Given: KLMN is a trapezoid m∠N = m∠KML ME⊥ KN ME = 3√5 KE = 8 LM/KN = 3/5 Since ME is⊥to KN∠KEM and ∠MEN =90° Since ∠KEM=90°then ΔKEM is a right triangle Since ΔKEM is a right trianglethen we can find KM using the Pythagorean Theorem Since ME=3√5 and KE=8 KM^2=ME^2+KE^2 KM^2=(3√5)^2+(8)^2 KM^2=45+64 KM^2=109 KM=√109 (√109≈10.44) Since KLMN is a trapezoid LM and KN must be parallel So ∠LMK = ∠MKE because they are alt. int. angles And since ∠N and ∠LMK are congruent (Given) and ∠LMK = ∠MKE so ∠MKE = ∠N Meaning ΔKMN is an isosceles triangle so MK = MN Since KM =√109 then MN also = √109 and because MN = √109 and ME = 8  the Pythagorean Theorem can be used (or since the same numbers are being used then it can be assumed that the ME=EN) EN²+ME²=MN² EN²+(3√5)²=(√109)² EN²+45=109 EN²=109-45 EN²=64 EN=√64 EN=8 Since KE+EN=KN 8+8=KN 16=KN Since LM/KN=3/5 and KN=16 6/5=3.2 so 3*3.2=LM LM=9.6 Since KLMN is a trapezoid 1/2(LM+KN)*ME=A (A=area) 1/2(9.6+16)*3√5=A 1/2(25.6)*3√5=A 12.8*3√5=A √7372.8=A