Question

Find the value of x in the figure below. Show all steps.

Answer 1

@mathmate

Answer 2

please help :c

Answer 3

Hint:
There is a very good article on the subject you're studying for learning or for reference:
http://www.regentsprep.org/regents/math/geometry/gp15/circleangles.htm

Look up the part on "two secants" and your answer is almost there!

Answer 4

no i still dont understand i just need ur help to solve this problem and be done with it, this is my last question

Answer 5

@mrnood plzzzzzzzzz

Answer 6

@phi

Answer 7

the angle labeled 9x-5 is equal to 1/2 the difference of the "intercepted" arcs.
First step: what is the difference between the intercepted arcs? (that means subtract the short arc from the bigger arc). what do you get ?

Answer 8

158 - 64 = 94

Answer 9

now divide it by 2, that gives the angle

Answer 10

94 /2 = 47

Answer 11

now you can say
9x-5= 47

Answer 12

to solve for x , first add 5 to both sides

Answer 13

9x = 47 + 5 = 52
9x = 52
x = 5.77

Answer 14

x= 52/9
if you change to a decimal it is 5.777....
the 7's repeat.
If you stop at the 2nd decimal you should round to 5.78 (because 5.777 is close to 5.780 than to 5.770)

But it might be better to leave it a fraction

Answer 15

oh ok i'll just go with 5.78 then? bcs im supposed to show all my work

Answer 16

I would leave it x=52/9
that is an exact answer
5.78 is approximate.

Answer 17

oh ok then thanks so much :)

Answer 18

OK - @phi @mathmate
This is quite an interesting problem

Mathmate gave a link to an almost total solution
phi gave a formula with no explanation

So the OP has the answer - but cannot show her work because no analysis has been done.

Can anyone derive the formula
"the angle labeled 9x-5 is equal to 1/2 the difference of the "intercepted" arcs" given by @phi from the geometry of the figure?

I am working on it - but not got time to finish it right now

a challenge if anyone (obviously not the OP who wants to 'be done with it') would like to come up with a derivation......????

Answer 19

Well, from the diagram, we see that triangles formed by radii are isosceles, so denoted all base angles x,y,p,q, etc.
From sum of angles of triangle,
c+p+x+q+x = 180
c+(180-p-y)+(180-q-y) = 180
Add both equations and simplify
c= y-x = \(\dfrac{180-b}{2}-\dfrac{180-a}{2}\)=\(\dfrac{a-b}{2}\) as required.
However, this may not be the shortest or the most direct proof. Please improve.

Answer 20

fine for me - well done