# How To Find Tangent Angle Of A Circle

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My problem has two different parts. I made the following diagram to illustrate the context. My questions are:

## How To Find Tangent Angle Of A Circle

Thank you! My geometry is a bit rusty and I would really like to figure out how to solve this problem.

#### Solved: ‘a, B And C Are Points On A Circle. Ef Is A Tangent To The Circle At C. D Is A Point On Ac. Angle Cbd:angle Abd = 3:1. Find Angle

So $;Delta BEC;$ is an isosceles triangle with sides $;8, 8, 4sqrt;$ and now you can use simple geometry to answer your question (more hint: $; angle EBC;$ is an exterior angle to $;Delta ABE;$ …)

$EB$ and $BC$ have equal lengths, so from your data, small $triangle ABE$ has hypotenuse of length $8$, short leg of length $2$, and you can find the length of $AE$ by Pythagoras theorem. Once you know the sides of the small triangle, you can find its angles in a minute. Finally, $angle EBC$ is $pi- angle ABE$.

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#### Question Video: Finding The Measure Of An Angle Given The Measure Of An Arc By Using The Properties Of Tangents To The Circle

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I marked two circuits as $A$ and $B$. Each of these circles has known positions $vec$ and $vec$ with radii $r_A$ and $r_B$ . I need to find the angle ($theta_A$ or $theta_B$) shown as the blue lines extending from the origin of the circles in the diagram. These theta angles correspond to the angle from the origin of the circles to the point where the tangent intersects the circle. The tangent must follow the diagram above, so that $theta_A = theta_B + 180$. We must also assume that these circles never intersect so that $d = lVert vec – vec rVert > r_A + r_B$.

Grow circle $A$ to a radius of $r_A+r_B$ while reducing that of $B$ to $0$. The common tangent keeps the same direction. So you have a right triangle with sides $P_AP_B$ and $r_A+r_B$. Now you add the direction angle of $P_AP_B$ and the angle of the triangle.

#### Find The Angle Between Pair Of Tangents Drawn From (1,3) To Circle X^2+y^2 2x+4y 11=0

Let us call the intersection of the line between the midpoints with the common tangent $O$. Then you get two similar right triangles. You can also write $$|costheta_A|=frac-vec||}=|costheta_B|=frac-vec||}$$ You also have that $$||vec- vec ||+||vec-vec||=||vec-vec||=d$$ From this you can get $$|costheta_A|=|costheta_B|= frac $ $ From $d>r_A+r_B$, get $|costheta_A|<1$

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#### Intersecting Secants Theorem (explained W/ 15 Examples!)

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