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In the figure, ABCD and BEFG are squares, and BCE  is an equilateral triangle. What is the number of degrees in angle GCE ?

 

 

tertre  Apr 18, 2017
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6+0 Answers

 #1
avatar+79819 
+2

Draw a circle with B  as a center and BA as a radius....

 

This circle will pass through the points  A, C, E  and G

 

And angle EBG  will  be a central angle that will intercept the minor arc  EG

And the measure of this angle = 90°

 

And angle  GCE  will be an inscribed angle  intercepting the same arc as  angle EBG 

 

And an inscribed angle intercepting the same arc as a central angle will have 1/2 the measure of that angle......so.....  angle GCE  =  (1/2) (90°)  =  45°

 

 

cool cool cool

CPhill  Apr 18, 2017
edited by CPhill  Apr 18, 2017
 #2
avatar+5552 
+2

Wow, that's really good!

I would not have thought to draw a circle ! smiley

Here is a picture, I thought I might as well share it:

 

hectictar  Apr 18, 2017
 #3
avatar+79819 
+1

 

Thanks for that pic, hectictar........your illustrations bring "life" to these answers....!!!

 

 

cool cool cool

CPhill  Apr 18, 2017
 #4
avatar+79819 
+2

 

 

Not to beat a dead horse, but.....here's a way easier solution based on hectictar's pic.....no circle drawing is actually required!!!!

 

Note that triangle CBG is isosceles with angle  CBG  = 150

Since CB  = BG.....so...  angle BCG  =  [ 180 - 150] / 2   =  15

 

But angle BCE  =  60  =   angle BCG + angle GCE

 

So

 

60  = 15  + angle GCE  

45  =  angle GCE 

 

 

cool cool cool

CPhill  Apr 18, 2017
edited by CPhill  Apr 18, 2017
 #5
avatar+5552 
+2

Oh yeah! I see that now too..

 

I really like the circle solution though.. it's very clever! smiley

hectictar  Apr 18, 2017
 #6
avatar+18777 
+1

In the figure, ABCD and BEFG are squares, and BCE  is an equilateral triangle.

What is the number of degrees in angle GCE ?

 

 

Center of a circle in B with radius r=BG=BE=BC

 

Central Angle Theorem:

The angle at the centre is twice the angle at the circumference.

 

 

\(\beta = \frac{\alpha}{2}=\frac{90^{\circ}}{2}=45^{\circ}\)

 

laugh

heureka  Apr 18, 2017
edited by heureka  Apr 18, 2017

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