In the given rohmbus JKLM, MK and JL are the diagonals
MK = 30, NL = 13, Angle MKL = 41
From the properties of the diagonal of rohmbus :
Diagonals bisect the angles of a rhombus.
So, the diagonal MK bisect the angle JKL
i.e Angle MKL = Angle JKM
Angle MKL = 41 (given)
so, Angle JKM = 41 degrees
As the angle JKL is the sum of angle JKM and MKL
Angle JKL = Angle JKM + Angle MKL
Angle JKL = 41 + 41
Angle JKL = 82
Opposite angles of a rhombus are equal.
In the given rohmbus the opposite angles are :
(i) Angle JML and Angle JKL
(ii) Angle MJK and Angle MLK
as they are equal
So,
Angle JML = Angle JKL
we have Angle JKL = 82 (solve Above)
So, Angle JML = Angle JKL = 82 degree
Angle JML = 82 degree
Angle MJK = Angle MLK
Let they are equal to x
So, Angle MJK = Angle MLK =x
The sum of all angle in a rohmbus is equal to 360 degree
So, Angle MJK + Angl MLK + Angle JML + Angle JKL = 360
x + x + 82 + 82 = 360
2x + 164 = 360
2x = 360 - 164
2x = 196
x = 196/2
x = 98
Angle MJK = Angle MLK =x = 98 degree
Angle MLK = 98 degrees
Propertie of Rohmbus :
In a rhombus, diagonals bisect each other at right angles.
In the given rohmbus JKLM
The diagonals JL and MK are bisect each other at 90 degree
So, Angle MNL = Angle LNK = Angle KNJ = Angle JNM = 90 dgrees
Angle MNL = 90 degree
Now for angle KJL,
In traingle KNJ, we have Angle KNJ = 90 degree and Angle MKJ = 41
Sum of all angles in a triangle is equal to 180 degree
So, for triangle KNJ
Angle KNJ + Angle KJL + Angle MKJ = 180
90 + Angle KJL + 41 = 180
Angle KJL +131 = 180
Angle KJL = 180 - 131
Angle KJL = 49 degrees
Angle KJL = 49 degrees
Sides:
In a rhombus, diagonals bisect each other at right angles.
In the given rohmbus JKLM
Diagonal are MK and JL wich are bisect at N
So,
(i) JN = NL
(ii) MN = NK
(i) JN = NL
We have NL = 13
JN = NL = 13
JN = 13
Since the diagonal JL is the sum of the JN and NL
JL = JN + NL
JL = 13 + 13
JL = 26
(ii) MN = NK
here we have MK = 30
The diagonal MK is the sum of MN and NK
MK = MN + NK
30 = NK + NK ( MN = NK)
30 = 2(NK)
NK = 30/2
NK = 15
for the side KL
In triangle KNL, we have Angle KNL = 90 degrees, NK =15, NL =13
Apply the pythagoras theorem :
Hypotenuse² = Perpendicular² + Base²
here we have Hypotenuse KL, Base NL and Perpendicular NK
KL² = NK² + NL²
KL² = 15² + 13²
KL²=225 + 169
KL² =394
KL = 19.84
Answer :
a) NK = 15
b) JL = 26
c) KL = 19.84
d) m
e) m
f) m
g) m
h) m