Circle theorem question

I

ian goh

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Hello, I am stuck on this question where I can find y but don't understand how to find x and z, I am not sure which circle theorem rule to use any help would be appreciated thanks!
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Go here and read all the way through.

The property of a circle that you appear to have used already appears after the one that seem to be ignorant of at present (or you would have been able to complete the task), so, if you reach that particular property and still don't know how to proceed, then you will need to read up again; what you need is definitely on that web page.

I trust you know what the ticks on lines AB & BC mean and that you are also familiar with the properties of isosceles triangles?

Come back and let us know how you got on, please. It would also be helpful if you would outline how you eventually arrived at your solution(s); if anything you have attempted isn't correct we can offer further advice.
 
If you have a triangle inside a circle with one side of the triangle being a diameter, then what type of triangle do you have?
 
Dr.Peterson said:
What special kind of triangle is ABC?

What is the measure of (minor) arc CD?
Click to expand...
I believe the measure of minor arc CD is the angle (y = 40°) that Ian has already determined (given that he knows it's twice the angle subtended at the circumference by the same arc, I presume; how else would he get that without first finding "z"?).

It would seem to be the properties that Steven G & I have alluded to that he needs to explore: angles in a semi-circle & properties of triangles. :)
 
To ian goh, [imath]\Delta ADC[/imath] is a right triangle. How do we know that?
No you can find [imath]z^{\circ}[/imath]
[imath][/imath][imath][/imath]
 
The Highlander said:
I believe the measure of minor arc CD is the angle (y = 40°) that Ian has already determined (given that he knows it's twice the angle subtended at the circumference by the same arc, I presume; how else would he get that without first finding "z"?).
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You may be misreading the figure. It is arc AD whose measure is 40°, not CD. Arc CD, which can be found using angle y, is useful for finding angle z.

Or, of course, you could just use triangle ADC, as pka has pointed out.
 
Dr.Peterson said:
You may be misreading the figure. It is arc AD whose measure is 40°, not CD. Arc CD, which can be found using angle y, is useful for finding angle z.

Or, of course, you could just use triangle ADC, as pka has pointed out.
Click to expand...
No, I am not misreading the figure. I simply presumed (wrongly, ofc) that you were referring to minor arc AD (without checking back to the diagram) because that was how he had appeared to have arrived at his answer of 40° for "y" (without first getting "z"). Mea culpa! :oops:
I will endeavour to check your comments more carefully in future. :thumbup:
Your suggestion does, indeed, lead towards evaluation of angle y and the repeated hints about the angles in a semi-circle also allow for finding not only angle z (as pka suggests) but also angle x, except that he appears to be lacking the knowledge of a basic fact about isosceles triangles that would lead to the final solutions required.

Note to Ian goh: In addition to the website I mentioned already on the Properties of Circles, please also visit this website on the Properties of Triangles. Together, these sites give you all the information you need to solve your problem (and the revision won't do you any harm at all!). :)
 
Your answers are now correct. (Except for missing units! "X=45" is not the same as "X=45°". Mathematics is a language that requires a minimum level of precision.)

Were you asked to calculate the sum of the three angles? I am curious as to why you have done so; it is not pertinent to finding the individual sizes of x, y & z.

The facts (knowledge) you needed to find these angles is summarized below. You appeared to be familiar with point 1 (in the picture) but did not seem to be aware of points 2 & 3 when you came to us for assistance. Several hints were provided to lead you towards these facts and I trust that is how you eventually arrived at your answers?
 

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The Highlander said:
Your answers are now correct. (Except for missing units! "X=45" is not the same as "X=45°". Mathematics is a language that requires a minimum level of precision.)

Were you asked to calculate the sum of the three angles? I am curious as to why you have done so; it is not pertinent to finding the individual sizes of x, y & z.

The facts (knowledge) you needed to find these angles is summarized below. You appeared to be familiar with point 1 (in the picture) but did not seem to be aware of points 2 & 3 when you came to us for assistance. Several hints were provided to lead you towards these facts and I trust that is how you eventually arrived at your answers?
Click to expand...
Tato is not the OP! We don't yet know whether the OP has learned anything.

(I make that mistake sometimes, too.)
 
Dr.Peterson said:
Tato is not the OP! We don't yet know whether the OP has learned anything.

(I make that mistake sometimes, too.)
Click to expand...
Indeed! Thank you for pointing that out. I had seen Tato's name crop up elsewhere as both questioner & contributor and just assumed this was his thread.

Should his posts (with the answers) not have been deleted? :mad:

Thank you for bringing it to my attention, Dr P. :)
 
The Highlander said:
No, I am not misreading the figure. I simply presumed (wrongly, ofc) that you were referring to minor arc AD (without checking back to the diagram) because that was how he had appeared to have arrived at his answer of 40° for "y" (without first getting "z"). Mea culpa! :oops:
I will endeavour to check your comments more carefully in future. :thumbup:
Your suggestion does, indeed, lead towards evaluation of angle y and the repeated hints about the angles in a semi-circle also allow for finding not only angle z (as pka suggests) but also angle x, except that he appears to be lacking the knowledge of a basic fact about isosceles triangles that would lead to the final solutions required.

Note to Ian goh: In addition to the website I mentioned already on the Properties of Circles, please also visit this website on the Properties of Triangles. Together, these sites give you all the information you need to solve your problem (and the revision won't do you any harm at all!). :)
Click to expand...
I thought that y was angle at the centre so i times it by 2 from angle DCA thats how i got 40° Ill tried using the resources you sent me and I think i understand it now thanks :)
 

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