Ex10.3
16.In triangle ABC,A= 32 degree and B =53 degree,If the area of ABC is 40 (cm square),find the length of AB
22.In a circle,O is the centre and the radius is 18 cm,the chord AB substends an obyuse angle θ at the centre . The area of OAB is 81開方3cm square
a)find the θ
b)find the exact length of AB
c)the area of arc AB
23.ABCDE is a regular pentagon inscribed in a circle of radius 6cm,find
a)the area of ABCDE
b)the length of CD
c) the length of CF

Thx for helping!!

Solution

16.
> angle C=180-32-53=95 degree (angle sum of triangle)
>since area of triangle ABC=0.5(BC)(AC)sinC, so 0.5(BC)(AC)sin95=40. Therefore, (BC)(AC)=80/sin95
>By sine formula, BC/sin32=AC/sin53. Therefore, BC/AC=sin32/sin53
>After solving equation in blue and green, we have BC app.equals to 7.3 cm and AC app. equals to 11cm
>By sine formula, AB/sin95=BC/sin32. Therefore, AB=13.7cm(to 3 sig. fig.)

22.
a.) Applying area of triangle OAB=0.5(OA)(OB)sinθ. We have 0.5(18)(18)sinθ=81square root 3. Therefore θ=120 dregee(As θ is an obtuse angle)

b.) Applying cosine formula, AB2=OA2+OB2-2(OA)(OB)cosθ,
therefore, AB2=324+324-2( 324)cos120. So AB=18square root 3 cm

c.) I do not know whether you mean area of sector OAB or arc length AB.
For area of sector OAB, applying the formula for area of sector,we have area of sector OAB=339cm2(to 3 sig. fig.)
For arc length AB, applying the formula for the arc length, we have arc AB=37.7cm(to 3 sig. fig.)

23.
a.)Divide the pentagon into 5 equal triangles by drawing lines from the centre of the circle to each corner of the pentagon.
>angle substended at the certre in each triangle=360/5=72 degrees (angle at a point)
>area of each triangle=0.5(r)(r)sin72=(0.5)(6)(6)sin72
> area of the pentagon=(5)(0.5)(6)(6)sin72=85.6cm2(to 3 sig. fig.)

b.)Is CD the side of the pentagon?
If yes, then by cosine formula, CD2=OC2+OD2-2(OC)(OD)cos72,
therefore, CD2=36+36-2(36)cos72. So CD=7.05cm(to 3 sig. fig.)

c.)
>angle CDE=[(5-2)(180)]/5=108 degree (angle sum of polygon)
>by cosine formula, CE2=CD2+DE2-2(CD)(DE)cos108,
therefore, CE2=49.7+49.7-2(49.7)cos108. So CE=11.4cm(to 3 sig. fig.)

Hope it can help you!