Geometry Question? Quadrilateral EFGH is inscribed in a circle. Sove for x. Then find the measure.. rest below

Question

...of angle H. Angle E= 9(x+7) Angle F= 6(x+12) Angle G= 5(21-x) If you could explain step by step that would be great. Thanks!

Answer 1

A quad inscribed in a circle is a square.
Opposite angles of any quad in a circle must be supplementary.

angle E supp angle G

9(x+7) + 5(21 - x) = 180

9x + 63 + 105 - 5x = 180

4x + 168 = 180

4x = 12
x = 3

Each angle = 90.

Answer 2

Given a concyclic quadrilateral EFGH, solve for x.

Angle E = 9(x + 7) = 9x + 63
Angle F = 6(x + 12) = 6x + 72
Angle G = 5(21 - x) = 105 - 5x

Opposite angles in a concyclic quadrilateral are supplementary. Therefore:

E + G = 9x + 63 + 105 - 5x = 180
4x + 168 = 180
4x = 180 - 168 = 12
x = 3°

So

Angle E = 9x + 63 = 9*3 + 63 = 27 + 63 = 90°
Angle F = 6x + 72 = 6*3 + 72 = 18 + 72 = 90°
Angle G = 105 - 5x = 105 - 5*3 = 105 - 15 = 90°

H = 180 - F = 180 - 90 = 90°

All four angles are 90°. Therefore quadrilateral EFGH is a rectangle. It is not necessarily a square.

Answer 3

Since you have two variables in the problem (x and H), and you are solving for H, you will still have x as a variable (just warning you).

9(x+7)+6(x+12)+5(21-x)+H=360
9x+6x+5x+63+72-105+H=360
20x+30+H=360
H=(20x+330) degrees
Thats the best I can do w/ the info given.

P.S. BTW I added this when I saw the reply above... It does not have to be a square. Quadrilateral merely means four sided shape. You may put the first three lines in any direction in the cirlcle as long as none intersect and the fourth connects the first to the third.

Answer 4

Actually, a quadrilateral inscribed in a circle does not have to be a square...

It could be a trapezoid or any type of parallelogram that is not a square, and it could just be a quadrilateral that is not special. All the quadrilateral has to do is have its sides all touching the circle.

This question is possible, and is a square, but I don't know how to prove it.

Answer 5

If we enable EFGH be a sq., then of course EF=FG=GH=HE and a=b=c=d. hence, this question does not uniquely % out a particular length of circle, except you recommend that it somewhat is actual for ANY possible values of a,b,c, and d. yet i will trivially arise with values of a,b,c, and d for which this does not be actual for a particular circle (enable a and d attitude 0, and b and c attitude 2r, as an example.) there is not any longer a different answer.