How do you find the angle between two tangents?

Tangent Troubles? Cracking the Code on Angles Between ‘Em

Tangents. Those straight lines that just kiss a curve at a single point. They’re more important than you might think! From designing bridges to making video games look good, understanding tangents is key. And one question that pops up a lot is: how do you figure out the angle when two of these tangents meet? Don’t worry, it’s not as scary as it sounds. Let’s break it down.

Circle Tangents from Way Outside

Okay, picture this: you’ve got a circle, and then you’ve got a point hanging out way outside the circle. Now, imagine drawing two tangent lines from that point to the circle. Where they meet at that outside point? That’s the angle we’re talking about, ∠APB (if we label things nicely).

Here’s the cool part: there’s a neat relationship at play. The angle where the tangents meet outside the circle and the angle formed by connecting the tangent points to the center of the circle always add up to 180 degrees. They’re supplementary, in math speak.

Think of it like this: if you draw lines from the circle’s center to where the tangents touch, you get two right angles (90° each). And since everything inside that four-sided shape has to add up to 360°, the other two angles have to make up the difference to reach 360 degrees which is 180 degrees.

So, how do you actually find the angle ∠APB?

Quick example: Let’s say ∠AOB is a chunky 110°. That means ∠APB is 180° – 110°, which is a neat 70°. Not so bad, right?

The Fancy Formula (When You Need It)

Sometimes, you don’t know the angle at the center. Instead, you might know the circle’s radius (r) and the length of the tangent line from the outside point to the circle (L). In that case, you can use this formula:

2θ = tan-1(2 * r * L / (L2 – r2))

Yeah, it looks a bit hairy, but it’s just plugging in numbers. It all comes from basic trigonometry and the right triangles you can draw in the picture.

How to use it:

Slopes to the Rescue: When You Know the Lines

What if you don’t have a circle at all, but you do know the equations of the two tangent lines? Or, at least, you can figure out their slopes (gradients)? Then you’re in luck!

If m1 and m2 are the slopes of the two lines, the angle θ between them is:

tan θ = |(m1 – m2) / (1 + m1 * m2)|

The vertical bars mean you take the absolute value (make sure the answer is positive).

Here’s how it works:

Two Circles, Double the Fun (or Trouble!)

Things get trickier when you’re dealing with tangents to two circles. Now you have to worry about whether the tangents cross between the circles (transverse tangents) or stay on the same side (direct tangents). And whether the circles intersect or not! Let’s not dive too deep into that rabbit hole right now. In the special case where two circles are tangent to each other, the “angle” between them is considered to be 0 degrees.

The Big Ideas: Keep These in Mind

• Tangents are perpendicular: A tangent always makes a right angle with the radius at the point where it touches the circle. Super important!
• Supplementary angles: Two angles that add up to 180°. Keep an eye out for them!
• Exterior Angle Theorem: A fancy way to relate angles inside and outside triangles. Useful in some tangent problems.
• Alternate Segment Theorem: Another angle trick that can come in handy.

Summing It Up

Finding the angle between tangents might seem tough at first, but it all boils down to understanding the relationships between lines, circles, and angles. Whether you’re using the 180° trick, a trig formula, or the slopes of lines, you’ve now got the tools to tackle those tangent troubles head-on! Go get ’em!