(a)
Since ABCDEFGH is a regular octagon, each interior angle is 135º.
Angle CDE = 180º - 135º = 45º.
Angle CDE = 135º + 45º = 180º.
(b)
(i)
Angle ODM is an exterior angle of triangle ODM, so angle ODM = 135º.
(ii)
Angle DOM = 360º - Angle ODM = 360º - 135º = 225º.
(c)
In triangle OMD,
cos(45º) = MD/OM
MD = OM * cos(45º)
MD = OM * (sqrt(2)/2)
(d)
(i)
Since triangle DOM is isosceles, the area of the triangle will be (1/2) * OM * MD
= (1/2) * OM * OM * (sqrt(2)/2)
= (OM^2 * sqrt(2))/4
(ii)
The area of the octagon can be calculated as the sum of areas of 8 triangles formed by joining the center O to each corner of the octagon.
Area of octagon = 8 * (1/2) * OM * OM * (sqrt(2)/2) = 4 * OM^2 * sqrt(2)
(e)
The area of the circle = π * r^2 = π * 10^2 = 100π cm^2.
ABCDEFGH is a regular octagon.
The circle, centre O, touches the sides of the octagon at their mid-points.
( a ) Show that angle CDE is 135º .
[3]
( b ) Find
( i ) angle ODM,
[1]
( ii ) angle DOM,
[1]
( c ) Using trigonometry, calculate the length of MD.
[3]
( d ) Calculate
( i ) The area of the triangle DOM,
[ 2 ]
( ii ) The area of the octagon.
[ 1 ]
( e ) Calculate the area of the circle, radius 10 cm.
1 answer