Please help asap

H

homeschool girl

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Question:
Triangle [MATH]WXY[/MATH] has side lengths [MATH]XY = 14[/MATH] and [MATH]WX = 8[/MATH]. The tangent to the circumcircle of triangle [MATH]WXY[/MATH] at [MATH]X[/MATH] is drawn, and the line through [MATH]W[/MATH] that is parallel to this tangent intersects [MATH]\overline{XY}[/MATH] at [MATH]Z.[/MATH] Find [MATH]YZ.[/MATH]
Screen Shot 2020-12-18 at 6.10.32 PM.png


I'm pretty sure that I have to use similar triangles, but I can't find equal angles that would help.
I would appreciate some guidance, thanks.
 
can I please have help?

pretty sure I have to use similar triangles and circle arcs but I don't know what to do next
 
oh, I think I get what I did wrong; I wrote question when I probably should've worded it differently sense there was no actual question in the problem
 
homeschool girl said:
Question:
Triangle [MATH]WXY[/MATH] has side lengths [MATH]XY = 14[/MATH] and [MATH]WX = 8[/MATH]. The tangent to the circumcircle of triangle [MATH]WXY[/MATH] at [MATH]X[/MATH] is drawn, and the line through [MATH]W[/MATH] that is parallel to this tangent intersects [MATH]\overline{XY}[/MATH] at [MATH]Z.[/MATH] Find [MATH]YZ.[/MATH]
View attachment 23853


I'm pretty sure that I have to use similar triangles, but I can't find equal angles that would help.
I would appreciate some guidance, thanks.
Click to expand...
There is, in fact, a pair of similar triangles, though I had to play with an accurate construction to see it. I should have seen it earlier, though.

Do you see any angles that are equal to XWZ? If not, what have you learned about angles inscribed in a circle?

It's a very nice problem! And you did ask (implicitly) a perfectly good question.
 
ohh, I read the link and I think I get it now, it would mean angle XWZ is equal to angle WZX, which would mean line ZY is equal to 6, right?
 
homeschool girl said:
this is what I got. I'm not sure where I went wrong.


View attachment 23857
Click to expand...
But b is not equal to a, because Z is not on the circle, as I said.

Look for an inscribed angle that is subtended by the same arc WX -- a point on the circle, not Z.

EDIT: It's hard to be sure of your reasoning, because you didn't label most points in this picture. If the line is WZ, parallel to the tangent, then your a's are equal, and your b's are equal, but the arc you labeled 2a does not subtend angle b.
 
Last edited:
Homeschool girl, You have angles labeled \(\alpha\) but they are not the same measure.
Let's add two points \(A~\&~B\) such that \(B-X-A\) such that \(B\) is closest to \(W\).
Now use the link provided by Prof. Peterson to see these.
\(m(\angle WXB)=m(\angle WYX)\)
\(m(\angle YXA)=m(\angle XWY)\)
\(m(\angle WZX)=m(\angle ZXA)\) (alternate interior angles)
 
I need to apologize as you did ask a valid question. Sorry about that. I read your post three times and did not see that at the end you had Find YZ.
Again, I apologize for that!
 
Beer soaked speculation follows.
homeschool girl said:
Question:
Triangle [MATH]WXY[/MATH] has side lengths [MATH]XY = 14[/MATH] and [MATH]WX = 8[/MATH]. The tangent to the circumcircle of triangle [MATH]WXY[/MATH] at [MATH]X[/MATH] is drawn, and the line through [MATH]W[/MATH] that is parallel to this tangent intersects [MATH]\overline{XY}[/MATH] at [MATH]Z.[/MATH] Find [MATH]YZ.[/MATH]
View attachment 23853


I'm pretty sure that I have to use similar triangles, but I can't find equal angles that would help.
I would appreciate some guidance, thanks.
Click to expand...
Using coordinate geometry and questionable speculation,
I get the impression that YZ = 66/7 or roughly 9.429
I googled one of homeschool girl's earlier questions a few weeks ago and it brought me to another math site.
When I googled homeschool girl's question again, it brought me to the same math site.
One of the experts there seems to agree with what I got but he used the tan function instead.
A moderator there also used coordinate geometry but also used different vertices.
web2.0calc.com

Can anyone solve this?

Triangle WXY has side lengths XY=14 and WX=8 . The tangent to the circumcircle of triangle WXY at X is drawn, and the line through W that is parallel to this tangent intersects XY at Z. Find YZ.
web2.0calc.com web2.0calc.com
web2.0calc.com

please help!

triangle WXY has side lengths XY = 14 and WX = 8. the tangent to the circumcircle of traingle WXY at X is drawn, and the line through W that is parallel to this tangent intersects line WY at Z. find YZ. https://latex.artofproblemsolving.com/e/c/a/eca24d79527b78ac58748594bc0f89585248cc1a.png
web2.0calc.com web2.0calc.com
web2.0calc.com

Help please! Thx! :)

1. A circle centered at A with radius 10 is externally tangent to a circle centered at B with radius 7. A line that is externally tangent to both circles intersects line AB at C. Find the length BC. 3.
web2.0calc.com web2.0calc.com
web2.0calc.com

help asap!

Triangle WXY has side lengths XY = 14 and WX = 8. The tangent to the circumcircle of triangle WXY at X is drawn, and the line through W that is parallel to this tangent intersects XY at Z. Find YZ.
web2.0calc.com web2.0calc.com
 
Last edited:
jonah2.0 said:
Using coordinate geometry and questionable speculation,
I get the impression that YZ = 66/7 or roughly 9.429
I googled one of homeschool girl's earlier questions a few weeks ago and it brought me to another math site.
When I googled homeschool girl's question again, it brought me to the same math site.
One of the experts there seems to agree with what I got but he used the tan function instead.
A moderator there also used coordinate geometry but also used different vertices.
Click to expand...
None of that is necessary; similar triangles are the way to go, and you get an exact answer. pka pointed the way in post #12; my hints likewise lead to the realization that triangle WXY is similar to ZXW, from which you can easily find the answer.
 
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