Common internal tangent

[kaloyankolev]

Hi,
I'm solving one problem and I can't understand how I must solve the second part of it.

We have two circles k₁(O₁;r₁) and k₂(O₂;r₂) which are external to each other (r₁ + r₂ < O₁O₂). The line t is their common internal tangent, which meets k₁ at point P and k₂ at point Q. The segment PQ crosses O₁O₂ at O.

• r₁ = r₂, O₁O₂ = 4r₁; angle POO₁ = ?
• r₁ = 2r₂, O₁O₂ = 6r₂; angle POO₁ = ?

I will upload the drawings which I made. I can't solve the second part. Can anyone give me some help?

https://ibb.co/wRnC7nn
https://ibb.co/McChSzm

 

[Dr.Peterson]

Hi,
I'm solving one problem and I can't understand how I must solve the second part of it.

We have two circles k₁(O₁;r₁) and k₂(O₂;r₂) which are external to each other (r₁ + r₂ < O₁O₂). The line t is their common internal tangent, which meets k₁ at point P and k₂ at point Q. The segment PQ crosses O₁O₂ at O.

• r₁ = r₂, O₁O₂ = 4r₁; angle POO₁ = ?
• r₁ = 2r₂, O₁O₂ = 6r₂; angle POO₁ = ?

I will upload the drawings which I made. I can't solve the second part. Can anyone give me some help?

https://ibb.co/wRnC7nn
https://ibb.co/McChSzm

The second is not very different from the first. Do you see that triangles OPO₁ and OQO₂ are similar? From that you can determine OO₁ and OO₂, and find the angles.

 

[kaloyankolev]

The second is not very different from the first. Do you see that triangles OPO₁ and OQO₂ are similar? From that you can determine OO₁ and OO₂, and find the angles.

I don't know how to show that the triangles are similar. I only know congruent triangles. Can you explain me how can I show that?

 

[Dr.Peterson]

I don't know how to show that the triangles are similar. I only know congruent triangles. Can you explain me how can I show that?

I can't imagine being given this problem before learning about similar triangles. What have you learned?

But you can search for information about similar triangles. All you need is to show that two pairs of corresponding angles are congruent (called AA). Then you can conclude that sides are proportional.

 

[kaloyankolev]

I can't imagine being given this problem before learning about similar triangles. What have you learned?

But you can search for information about similar triangles. All you need is to show that two pairs of corresponding angles are congruent (called AA). Then you can conclude that sides are proportional.

I don't understand when I show that two pairs of corresponding angles are congruent what I have to do? I have already made that:

• angle O₁ = angle O₂
• angle POO₁ = angle QOO₂

Can you explain me what I have to get from the "similar triangles" as you said?

 

[Dr.Peterson]

I don't understand when I show that two pairs of corresponding angles are congruent what I have to do? I have already made that:

• angle O₁ = angle O₂
• angle POO₁ = angle QOO₂

Can you explain me what I have to get from the "similar triangles" as you said?

Have you made the effort to learn about similar triangles yet? You must do so.

Here is one source I found by searching for the phrase: https://www.mathsisfun.com/geometry/triangles-similar.html

You will be writing a proportion.