Question

THe interior angles of an n-sided convex polygon are in AP. If the common difference is 5 and the largest angle is 160, then n equals? @ganeshie8

Answer 1

lets try and setup two equations in "n" and "a"

Answer 2

largest angle is 160 :\[a+(n-1)5 = 160\tag{1}\]

Answer 3

interior angles add up to 180(n-2) :
\[\dfrac{n}{2}\left[2a+(n-1)5\right] = 180(n-2)\tag{2}\]

Answer 4

two equations and two unknowns, you can solve them

Answer 5

one sec

Answer 6

its a bit trickier than the previous problem as you can see the second equation is quadratic, so you will need to do a bit more work

Answer 7

cant get it

Answer 8

show me

Answer 9

try to substitute the value of "a" from first equation into second equation

Answer 10

i tried but it doesnt seem to work

Answer 11

wait

Answer 12

ok waiting

Answer 13

is it 16

Answer 14

@ganeshie8

Answer 15

doesn't look correct

Answer 16

isolating "a" from first eqn gives you
\[a = 160-5(n-1)\]

substitute that in second eqn

Answer 17

\[\dfrac{n}{2}\left[2(160-5(n-1))+(n-1)5\right] = 180(n-2)\]

simplify a bit and you will see a nice quadratic in "n"

Answer 18

what is it

Answer 19

i cant seem to get it

Answer 20

have you understood everythign so far ?

Answer 21

yes

Answer 22

Okay do you know to use distributive property and combine like terms ?

Answer 23

yes

Answer 24

i just cant gettit for some strange reason

Answer 25

then whats stopping you from simplifying the above mess ?

Answer 26

\[\dfrac{n}{2}\left[2(160-5(n-1))+(n-1)5\right] = 180(n-2)\]

start by distrubuting

\[\dfrac{n}{2}\left[2\times 160-2\times 5(n-1))+5(n-1)\right] = 180\times n-180\times 2\]

Answer 27

i know
i didnt get a quadratic but got n=18 which is wrong

Answer 28

\[\dfrac{n}{2}\left[2\times 160-5(n-1)\right] = 180n-360\]

Answer 29

\[\dfrac{n}{2}\left[325-5n\right] = 180n-360\]

Answer 30

multiply 2 both sides

\[n\left[325-5n\right] = 360n-720\]

Answer 31

the roots are -16 and 9

Answer 32

so its 9 oooops forgot to redo the minus

Answer 33

Yep! discard -16 as the number of sides cannot be negative

Answer 34

The key part is setting up the equations correctly using the properties of AP and triangles.
Once you have the equations, you can solve them using any of your favorite methods