Photo AI

The diagram shows a hexagon ABCDEF - Edexcel - GCSE Maths - Question 22 - 2017 - Paper 1

Question 22

The diagram shows a hexagon ABCDEF. ABEF and CBED are congruent parallelograms where AB = BC = cm. P is the point on AF and Q is the point on CD such that BP = BQ =... show full transcript

Worked Solution & Example Answer:The diagram shows a hexagon ABCDEF - Edexcel - GCSE Maths - Question 22 - 2017 - Paper 1

Step 1

Prove that cos PBQ = -√3/2

96%

114 rated

Answer

Since angle ABC = 30°, we can write

$ext{cos} 30° = rac{ ext{adjacent}}{ ext{hypotenuse}} = rac{BP}{AB}$

So,

$ext{cos} 30° = rac{BP}{10}$

This gives us

$BP = 10 imes rac{ ext{\sqrt}3}{2} = 5 ext{\sqrt}3$ .

Step 2

for P on AF and Q on CD, derive the equation

99%

104 rated

Answer

Using the properties of the congruent parallelograms, we know that:

$AB = BC = x ext{ cm}$

And since BP = BQ = 10 cm, we can relate the lengths:

$PQ = QP = BP + BQ = 10 + 10 = 20 ext{ cm}$ .

Step 3

for cos PBQ = 1 - (2 - √3)² / 200

96%

101 rated

Answer

By substituting the known values and rearranging:

$ext{cos} PBQ = rac{10^2 - 20^2 + (2 - ext{√}3)^2}{2 imes 10 imes 10}$

Expanding this leads us to:

$ext{cos} PBQ = rac{100 - 400 + 4 - 4 ext{\sqrt}3}{200}$ .

Step 4

Conclusion of proof

98%

120 rated

Answer

Thus,

$ext{cos} PBQ = rac{1 - (2 - ext{√}3)^2}{200}$

which completes the proof.

Join the GCSE students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;