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# Determining tangent lines: lengths

Solve two problems that apply properties of tangents to determine if a line is tangent to a circle.

## Problem 1

Segment start overline, O, C, end overline is a radius of circle O.
A circle centered around point O. Segment O C is a radius of the circle. Point A lies outside the circle, and line A C is a line that could potentially be tangent to circle O. A line segment connects point A to point O and intersects the circle at point B. Line segment A O, line segment O C, and line A C create the triangle A O C. Side A C of the triangle is eleven units. Side O C of the triangle is five units. Side A O is broken into two line segments, A B and B O. Line segment A B is eight units. Line segment B O is unknown.
Note: Figure not necessarily drawn to scale.
Is line A, C, with, \overleftrightarrow, on top tangent to circle O?

## Problem 2

Segment start overline, O, C, end overline is a radius of circle O.
A circle centered around point O. Segment O C is a radius of the circle. Point A lies outside the circle, and line A C is a line that could potentially be tangent to circle O. A line segment connects point A to point O and intersects the circle at point B. Line segment A O, line segment O C, and line A C create the triangle A O C. Side A C of the triangle is sixteen units. Side O C of the triangle is twelve units. Side A O is broken into two line segments, A B and B O. Line segment A B is eight units. Line segment B O is unknown.
Note: Figure not necessarily drawn to scale.
Is line A, C, with, \overleftrightarrow, on top tangent to circle O?

## Want to join the conversation?

• how is angle AOC not a right angled triangle in problem 1
(1 vote)
• The the first example is not a right triangle because it does not follow the Pythagorean Theorem of a^2 + b^2 = c^2. AC^2+OC^2 doesn't equal AO^2. (11^2 + 5^2 = 13^2, which turns out to be 146 = 169, not true). Both 45-45-90 and 30-60-90 triangles follow this rule.
• The first question is vague and doesn't explain how they found the length of AO. Where did y'all even get 8?
• 8 was given as the length of AB. BO is a radius of the circle and therefore has length of 5.
So, AO (which equals AB + BO) is 8 + 5 which is 13.
They also had to test whether there was a right angle at point C, so they used the Pythagorean theorem to see whether that was true. It wasn't, so in other words, AC wasn't perpendicular to OC, so therefore line AC was NOT tangent to circle O.
• how can we draw 2 common transverse tangents for 2 congruent circles if they have any distance between their centres?
• you dont that is something different you are using Pythagorean theorem here.
• how many types of tangent
• Can someone explan #2 to me?
I dont understand because 2,3,4 is not a right triangle??
• The sides of the triangle in problem 2 are 12, 16, and 20 (12+8), which does make it a right triangle, since 20² = 12²+16².
• Can someone explain why for problem two line BO is included in solving the problem while in problem 1 BO is left out? I understand that for problem 1 using the pythagorean theorem shows its not perpendicular but using that same method for problem 2 doesn't work and thus adding line BO is needed.
• You can find the length of BO in either question, using just the radius. This is what you use to find out if it is a right triangle and thus, you need BO. Hope this answers your question...
• what is the converse Pythagorean theorem?
(1 vote)
• Normally we use the Pythagorean Theorem on a Right Triangle to find the length of a missing side measurement.

•The Converse Pythagorean Theorem is when we already have the side lengths of a triangle, but we don't know if it's a Right Triangle

so it's when we use the formula to check if a triangle has a 90° angle through Comparison.

We must…

•Plug two of the triangle's side measurements into the corresponding variables of the Pythagorean Theorem.

•Calculate to see what the third side would be if it were a Right Triangle.

•Compare the triangle's given measurement to the calculated.

•If the corresponding values are equal then it's a Right Triangle and has a 90° angle.
If not, it is not.

So…
•To use the Pythagorean Theorem as its Converse, is to apply the formula to prove or disprove if a triangle has a 90° angle, and therefore is, or is not a Right Triangle.

(ㆁωㆁ) Hope this helps someone!