Calculate the QM and the radius

[chijioke]

Since you got a different answer, what you did is simply wrong. The question is, why?

First, this work accomplished nothing:

Yes because I ended from where I started from. I can see it clearly.

View attachment 39618​

Then, the following is just wrong; there's a major sign error:

View attachment 39619​

In fact, [imath]r^2-(15-r)^2=r^2-(225-30r+r^2)=30r-225[/imath]. So the next line should be

[imath]30r-225=64\\30r=289\\r=289/30\approx9.63[/imath]​

I remembered the identity
[math](a-b)^2=a^2-2ab+b^2[/math]I had to apply before I saw that
[math]30r = 64 +225[/math]Thanks for helping my memory.

 

[chijioke]

For the sake of completeness here is another, more geometric, approach: draw OH orthogonal to QR and use triangle similarities:
View attachment 39621

I showed something like this on #12.
But my challenge with these one is that I will not know that the triangles chosen are similar until I will have finished my work and compare the ratios to see if they are same then I can agree that my work is correct.

 

[Dr.Peterson]

I will not know that the triangles chosen are similar

Have you tried looking for similar triangles in his figure? You CAN know that they are similar before using them; in fact, you MUST.

Typically, the way to recognize similar triangles is to mark angles in the figure, and find triangles in which the three angles are the same.


Which angles are right angles? Mark them.

Then, what other angle is congruent to angle OQH? Mark those congruent angles as, say, x. (To see this, you'll want to first mark two segments as r, and notice two congruent triangles. It took me a while to see this in the figure, which is not exact.)

Now, what three triangles all have a right angle and an angle x? Those are similar, because triangles with two pairs of congruent angles are similar.

Then you can try using ratios to solve the problem.

Of course. you could also look back at your earlier question and try to learn from our answers.

 

[chijioke]

Have you tried looking for similar triangles in his figure? You CAN know that they are similar before using them; in fact, you MUST.

Typically, the way to recognize similar triangles is to mark angles in the figure, and find triangles in which the three angles are the same.

View attachment 39641
Which angles are right angles? Mark them.

Then, what other angle is congruent to angle OQH? Mark those congruent angles as, say, x. (To see this, you'll want to first mark two segments as r, and notice two congruent triangles. It took me a while to see this in the figure, which is not exact.)

Now, what three triangles all have a right angle and an angle x? Those are similar, because triangles with two pairs of congruent angles are similar.

Then you can try using ratios to solve the problem.

Of course. you could also look back at your earlier question and try to learn from our answers.

I just showed that angle QNO = QGO = OGR = ONT = 90 degree in triangle because |NO| and |OG| are perpendicular bisectors of |QT| and |QR| respectively.

If the triangle QTR is inscribed into a circle, then |QO|=|TO|= |OR|= r = radius of the circle. Can I now conclude that these four pairs of angles are congruent?

Are congruent triangles always similar?

Are triangles TOM and OMR also congruent or similar?

If all the triangles mentioned are similar, how do I know the corresponding sides?

 

[Dr.Peterson]

I just showed that angle QNO = QGO = OGR = ONT = 90 degree in triangle because |NO| and |OG| are perpendicular bisectors of |QT| and |QR| respectively.

There is no need to introduce new points G and N. How do you think that helps?

If the triangle QTR is inscribed into a circle, then |QO|=|TO|= |OR|= r = radius of the circle. Can I now conclude that these four pairs of angles are congruent?

WHICH four pairs of angles? You listed three lengths!

Are congruent triangles always similar?

Yes. Why do you think it could be otherwise?

Are triangles TOM and OMR also congruent or similar?

They are neither, when named this way. They are congruent (and therefore also similar) under the names TOM and ROM, which indicates which vertices correspond.

If all the triangles mentioned are similar, how do I know the corresponding sides?

By first identifying the corresponding vertices,

But you have done none of what I suggested. Why not?

It looks like you need to go back and learn the definitions of "congruent" and "similar". Please do so.

 

[chijioke]

There is no need to introduce new points G and N. How do you think that helps?

Just for easy reference.

WHICH four pairs of angles? You listed three lengths!

Sorry, I mean to say triangles QNO, TNO, QHO and OHR.
I don't know if I am naming it according the corresponding vertices?

Yes. Why do you think it could be otherwise?

I just want to learn.

They are neither, when named this way. They are congruent (and therefore also similar) under the names TOM and ROM, which indicates which vertices correspond.

Okay but could you please post some links where triangles are named according their corresponding vertices. I just want to learn.

By first identifying the corresponding vertices,

But you have done none of what I suggested. Why not?

It looks like you need to go back and learn the definitions of "congruent" and "similar". Please do so.

Yes I need to please send me some links to some materials or site that can help in that regard.

 

[Dr.Peterson]

Just for easy reference.

But that doesn't make anything easier!

Sorry, I mean to say triangles QNO, TNO, QHO and OHR.
I don't know if I am naming it according the corresponding vertices?

No, triangles QNO and TNO are neither similar nor congruent, in any order; QHO and OHR are congruent in the order QHO and RHO.

Do you see why? Telling us why you think something is true or false would help us see how you might be thinking, so we could help more.

Yes I need to please send me some links to some materials or site that can help in that regard.

You can probably do the search yourself (for, say, "similar and congruent triangles", as I just did); but here are some on sites I like:

Congruent Triangles

Triangles are congruent when they have exactly the same three sides and exactly the same three angles. It means that one shape can become...

www.mathsisfun.com

Similar Triangles

Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around). These triangles are all similar:

www.mathsisfun.com

Congruent Triangles Outline - MathBitsNotebook (Geo)

mathbitsnotebook.com

Methods of Proving Triangle Congruent - MathBitsNotebook(Geo)

mathbitsnotebook.com

Similarity Outline - MathBitsNotebook (Geo)

mathbitsnotebook.com

Proving Similar Triangles - MathBitsNotebook(Geo)

mathbitsnotebook.com