Question

Topic 6: Kites (Gina Wilson, Geometry)

Answer 1

Answer:

35.

So that we have:

KLN = 54          JKN =36

LKN = 72          NMJ = 18

KNM = 126       JLM = 72

LJM = 90          KLM = 126

36.

CE = 6√13

37.

m∠Z = 88°

38.

m∠GFE = 85

Step-by-step explanation:

35.

As KLMN is a kite:

+) KL = KN

=> Triangle KLN is  an isosceles triangle.

=> m∠KLN = m∠ KNL = m∠KNJ = 54°

In triangle KLN, total measure of three internal angles are 180 degree, so that: m∠KLN + m∠ KNL + m∠ LKN = 180°

=> 54 + 54 + m∠ LKN = 180°

=> m∠ LKN = 180 - 54 - 54 = 72°

+) KM is the bisector of both Angle LKN and Angle LMN

So that we have:

• m∠JKN = 1/2 m∠LKN = 1/2 * 72 = 36°
• m∠NMJ = 1/2 m∠LMN = 1/2 * 36 = 18°

+) KM and LN is perpendicular to each other at point J

=> m∠LJM = 90°

+) m∠KLM = m∠KNM; m∠KLM + m∠KNM + m∠LKN + m∠LMN = 360°

=> m∠KLM + m∠KLM + 72 + 36 = 360

=> 2 m∠KLM = 360 - 72 - 36 = 252

=> m∠KLM = 252/2 = 126 = m∠KNM

+) We have: m∠KLM = m∠KLN + m∠JLM

=> m∠JLM = m∠KLM - m∠KLN = 126 - 54 = 72

36.

As BCDE is a kite, we have:

+) BC = BE; CD = DE = 21

+) BD is perpendicular to CE and intersect each other point F

=> Angle CFD = 90

=> Triangle CFD is the right triangle

According to Pythagoras theorem: CF^2 + DF^2 = CD^2

=> CF^2 = CD^2 - DF^2  = 21^2 - 18^2 = 441 - 324 = 117

=> CF = $\sqrt{117} =3\sqrt{13}$

+) According to the feature of a kite shape, F is midpoint of CE

=> CF = FE = 1/2 x CE

=>CE = 2 x CF = 2 x $3\sqrt{3} = 6 \sqrt{13}$

So that CE = 6√13

37.

As given, WXYZ is a kite shape.

According to the feature of a kite shape, angle ZWX and angle ZYX are equal to each other.

=> ∠ZWX = ∠ZYX

=> 8x -23 = 6x +11

=> 8x - 6x = 11 + 23

=> 2x = 34

=> x = 34/2 = 17

=> ∠ZWX = 8x - 23 = 8*17 - 23 = 113°

=> ∠ZWX = ∠ZYX = 113°

As WXYZ is a kite shape, so that total measure of 4 internal angles of it are 360 degree

So that we have:

∠Z + ∠ZWX + ∠ZYX + ∠WXY = 360

=> ∠Z + 113 + 113 + 46 = 360

=> ∠Z = 360 -113 -113-46 = 88°

So that the measure of ∠Z is 88°

38.

As GDEF is a kite shape. As we can see, GE is the longer diagonal, so that it is the bisector of both Angle DGF and Angle DEF.

H is on GE and GE is the bisector of Angle DEF, so we have:

m∠HEF = m∠GEF = 1/2. m∠DEF = 1/2 . (12x- 16)= 6x - 8

As GDEF is a kite shape so that the two diagonals DF and GE are perpendicular to each other, so that: m∠EHF = 90

In the triangle EHF, m∠EHF = 90

=> EHF is a right triangle

=> m∠HEF + m∠EFH = 90

=> 6x - 8 + 3x -1 = 90

=> 9x = 90 + 8 + 1 = 99

=> x = 99/9 =11

=> m∠HEF = 6x - 8 = 6*11 - 8 = 58 = m∠GEF

GE is the bisector of both Angle DGF

=> m∠EGF = 1/2. m∠DGF = 1/2 . 74 = 37

In the triangle GEF, total measure of three internal angles are 180

=> m∠GFE + m∠EGF + m∠GEF = 180

=> m∠GFE + 37 + 58 = 180

=> m∠GFE = 85