Respuesta :
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
Side KM is equal to 10.44 units while the area of trapezoid KLMN is equal to 85.87 square units.
How to calculate the length of sides.
In order to determine the length of side KM, we would apply Pythagorean's Theorem because angles MEN and KEM are right angles (90°).
Mathematically, Pythagorean Theorem is given by this formula:
c² = a² + b² ⇒ KM² = ME² + KE²
KM² = (3√5)² + 8²
KM² = 45 + 64
KM² = 109
KM = √109
KM = 10.44 units.
Next, we would determine side LM and KN:
Since trapezoid KLMN has equal alternate interior angles (∠LMK = ∠MKE), sides LM and KN must be parallel to each other. Therefore, we have:
KN = KE + EN
But, EN and KE = √64 = 8 units.
KN = 8 + 8
KN = 16 units.
For side LM, we have:
LM/KN = 3/5
LM/16 = 3/5
Cross-multiplying, we have:
5LM = 48
LM = 48/5
LM = 9.6 units.
How to calculate the area of a trapezoid.
Mathematically, the area of a trapezoid is given by this formula:
A = ½ × (a + b) × h ⇒ A = ½ × (LM + KN) × ME
A = ½ × (9.6 + 16) × 3√5
A = ½ × 25.6 × 3√5
A = 12.8 × 3√5
A = 85.87 square units.
Read more on trapezoid here: https://brainly.com/question/4758162
