You are given a sector which subtend an angle of 70 degree at the centre of circle with radius 21 cm find the length of arc and area of given sector
Answers
Answer:
here is your answer
area sector =θ/360×πr²
= 70/360×22/7×21×21
= 7/36×22×3×21
= 7/36×66×21
= 7/18×33×21
= 7/6×11×21
= 7/6×131
= 917/6cm²
area arc= θ/360×2πr
= 70/360×2×22/7×21
= 7/36×44×3
= 7/12×44
= 7/3×11
= 77/3
may it help
Answer:Here’s a sketch of the problem.
And we are given that ∠ACB=72∘
We are asked to find the length of the arc AB and the length of the chord AB
Notice that I have added the bisector of ∠ACB , the blue line through C and D.
The reason I did this, and the reason for my writing this answer when there already is one offered, is to plant the thought that there are two major ways of approaching geometry and math.
One way is to memorize a long list of rules and special purpose formulas and recognize when to apply them.
The other way is to realize that the special formulas were derived through some hack that was based upon a short list of more basic tools.
You have a choice.
Memorize the special formula and do enough problems so that you recognize exactly when the special formula is needed.
Or,
Remember the hack, and repeat the hack with the relevant details when it comes up.
Life is too short for most of us to memorize the formula for a chord given the angle and radius. And if we do memorize it for the test, that memory is going to get tossed out of your brain like the combination for your gym locker three years ago when you no longer need to know the length of a chord. So much for impressing your kids thirty years from now.
I say, build your muscle memory of the hack.
First, let’s look at the problem of the length of the arc. Here is the hack:
The angle swept through part way around the circle is to a full rotation around the circle as the length of the arc is to the length of the full circumference. The areas are also going to be in the same proportion, but we don’t need that fact right now.
All the way around the circle is 360∘ also known as [math]2 \pi \[/math] radians. So what fraction of the way around is 72∘ ?
72∘360∘=15
Since we remember the radians around a circle, let’s derive the radian equivalent.
Radian measure of ∠ACB=15⋅2π radians.
Radian measure of ∠ACB=2π5
And, since the arc has the same proportion with the whole circumference
ArcABCircumference=15
Arc AB=Circumference5
Circumference=2⋅π⋅r and we are given that r=21
So,
Arc AB=2⋅π⋅215=42π5 units.
Now, let’s hack the chord formula.
By definition, that extra line that bisects ∠ACB makes ∠ACD=∠DCB.
Our radii are the same length and the two triangles △ACD and △DCB share the side CD.
The two triangles are congruent and the sides opposite the bisected angles are thus the same length. The length of either one is half the length of our chord. The full length of the chord is twice the length AD.
Step-by-step explanation: