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QUESTION FOUR (8 marks)
There are four lakeside homes at A, B, C, and D on a man-made quadrilateral lake. There is also a circular road arour the lake that connects the four homes. (As seen in the diagram below). Each of the four families living in the homes argues that their house has the widest angled view of the lake.
It is known that:
i) Water pipes FA and CE run at a tangent to the circle road.
ii)
AF is parallel
BD.
iii) Lines
BD and
EF both run through the centre of the circle road at
O.
iv) Angle
/_AFO is
47^(@)
v) Angle
/_CEO is
32^(@)
Give geometric reasons for all your working.
a) Find the value of angle
/_FOB
b) Find the value of angle
/_OAF
c) Calculate which house A, B, C, or D has the widest angled view of the lake and the value of that angle.
d) Given that length AF is
1.5km, calculate the distance around the roas from house A to house C.

QUESTION FOUR ( $8$ marks) $\newline$ There are four lakeside homes at A, B, C, and D on a man-made quadrilateral lake. There is also a circular road arour the lake that connects the four homes. (As seen in the diagram below). Each of the four families living in the homes argues that their house has the widest angled view of the lake. $\newline$ It is known that: $\newline$ i) Water pipes FA and CE run at a tangent to the circle road. $\newline$ ii) $A F$ is parallel $B D$ . $\newline$ iii) Lines $B D$ and $E F$ both run through the centre of the circle road at $O$ . $\newline$ iv) Angle $\angle A F O$ is $47^{\circ}$ $\newline$ v) Angle $\angle \mathrm{CEO}$ is $32^{\circ}$ $\newline$ Give geometric reasons for all your working. $\newline$ a) Find the value of angle $\angle F O B$ $\newline$ b) Find the value of angle $B D$ $0$ $\newline$ c) Calculate which house A, B, C, or D has the widest angled view of the lake and the value of that angle. $\newline$ d) Given that length AF is $B D$ $1$ , calculate the distance around the roas from house A to house C.

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Q. QUESTION FOUR (

8

marks)

\newline

There are four lakeside homes at A, B, C, and D on a man-made quadrilateral lake. There is also a circular road arour the lake that connects the four homes. (As seen in the diagram below). Each of the four families living in the homes argues that their house has the widest angled view of the lake.

\newline

It is known that:

\newline

i) Water pipes FA and CE run at a tangent to the circle road.

\newline

ii)

A F

is parallel

B D

\newline

iii) Lines

B D

and

E F

both run through the centre of the circle road at

O

\newline

iv) Angle

\angle A F O

47^{\circ}

\newline

v) Angle

\angle \mathrm{CEO}

32^{\circ}

\newline

Give geometric reasons for all your working.

\newline

a) Find the value of angle

\angle F O B

\newline

b) Find the value of angle

B D

0

\newline

c) Calculate which house A, B, C, or D has the widest angled view of the lake and the value of that angle.

\newline

d) Given that length AF is

B D

1

, calculate the distance around the roas from house A to house C.

Angle Calculation: To find the value of angle $\angle FOB$ , we will use the fact that $AF$ is parallel to $BD$ and that lines $BD$ and $EF$ both run through the center of the circle road at $O$ . Since $AF$ is parallel to $BD$ and $FO$ is a transversal, angle $\angle AFO$ is congruent to angle $\angle FOB$ due to alternate interior angles in parallel lines. $\newline$ Angle $\angle AFO$ is given as $AF$ $2$ degrees. $\newline$ Therefore, angle $\angle FOB$ is also $AF$ $2$ degrees.
Angle Calculation: To find the value of angle $\angle OAF$ , we will use the fact that angle $\angle AFO$ is given as $47$ degrees and that angle $\angle OAF$ is the complement to angle $\angle AFO$ because they form a right angle at point F (since FA is tangent to the circle at point F). $\newline$ Angle $\angle OAF = 90$ degrees $-$ angle $\angle AFO$ $\newline$ Angle $\angle OAF = 90$ degrees $-$ $47$ degrees $\newline$ Angle $\angle AFO$ $1$ degrees
Widest Angle View: To calculate which house has the widest angled view of the lake, we need to consider the angles at the center of the circle road. Since $BD$ and $EF$ are both diameters of the circle and pass through the center $O$ , the angles at $A$ , $B$ , $C$ , and $D$ are central angles of the circle. $\newline$ The widest angle will be opposite the longest arc, which is the semicircle. $\newline$ Since angle $/_AFO$ is $47$ degrees and angle $/_CEO$ is $EF$ $0$ degrees, the remaining angle at $O$ (angle $EF$ $2$ ) is the sum of these two angles. $\newline$ Angle $EF$ $3$ $\newline$ Angle $EF$ $4$ degrees $EF$ $5$ degrees $\newline$ Angle $EF$ $6$ degrees $\newline$ The widest angle view is at house $D$ , which is opposite to angle $EF$ $2$ . $\newline$ Angle at house $EF$ $9$ degrees $O$ $0$ $\newline$ Angle at house $EF$ $9$ degrees $O$ $2$ degrees $\newline$ Angle at house $O$ $3$ degrees
Distance Calculation: To calculate the distance around the road from house A to house C, we need to find the circumference of the circle road. Since AF is a tangent to the circle and is given as $1.5 \, \text{km}$ , it is also the radius of the circle. $\newline$ The circumference of a circle is given by the formula $C = 2 \pi r$ , where $r$ is the radius. $\newline$ $C = 2 \pi \times 1.5 \, \text{km}$ $\newline$ $C = 3 \pi \, \text{km}$ $\newline$ Since we want the distance from A to C, we need half of the circumference, as it represents the semicircle from A to C. $\newline$ Distance from A to C = $\frac{1}{2} C$ $\newline$ Distance from A to C = $\frac{1}{2} \times 3 \pi \, \text{km}$ $\newline$ Distance from A to C = $1.5 \pi \, \text{km}$