Tangents and Circles

Circle Tangents and Radii

Just what is going on with these two?

When a tangent line and circle radius meet...

• If C is the center of a circle and B is a point on that circle, then 

  
  

  AB is tangent to ⨀C if and only if 

  
  

  AB is perpendicular to BC

  
  

What Is XZ?

Since YZ is tangent to ⨀X, YZ is perpendicular to XY. So, △XYZ is a right triangle with hypotenuse XZ. We can use PYTHAGOREAN THM to solve for the missing side.

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Solve for XZ

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ST is tangent to ⨀Q. What is QT?

Multiple Choice

GH is tangent to ⨀F. What is m∠H ?

• 30 degrees

• 20 degrees

• 35 degrees

• 90 degrees

In a right triangle, the non-right angles are complementary.

Since PT is tangent to ⨀O, △OPT is a right triangle with right angle ∠P. So, ∠O and ∠T are complementary. 

Thus 

 $m\angle O+m\angle T=90\degree$  

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TU is tangent to ⨀S. What is m∠U ?

Multiple Choice

VW is tangent to ⨀U. What is m∠X ?

• 35

• 90

• 45

• 55

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PQ is tangent to ⨀N. What is NP ?

tangent lines form right angles with circle radii

Since PQ is tangent to ⨀N, PQ is perpendicular to NP. So, △NPQ is a right triangle with hypotenuse NQ.

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MN is tangent to ⨀K. What is MN ?

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JK is tangent to ⨀H. What is m∠K ?

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XY is tangent to ⨀W. What is XY?

Is it a right triangle?

In order to determine if a triangle is a right triangle, we can use the CONVERSE OF THE PYTHAGOREAN THEOREM.

Multiple Choice

Converse of the Pythagorean Theorem: Is the following a right triangle?

• $$c^2=a^2+b^2$$ yes!

• $$c^2\ne a^2+b^2$$ no.

Multiple Choice

Determine if AB is tangent to the circle centered at point C. Explain your reasoning.

• $$\angle A$$ is a right angle by the converse of the Pythagorean Theorem, thus AB is perpendicular to the radius AC. So AB is tangent.

• $$\angle A$$ is a right angle by the converse of the Pythagorean Theorem, thus AB is perpendicular to the radius AC. So AB is Not tangent.

• $$\angle A$$ is NOT a right angle by the converse of the Pythagorean Theorem, thus AB is NOT perpendicular to the radius AC. So AB is Not tangent.

Multiple Choice

A point B is 17cm away from the centre O of a circle. AB is the tangent to the circle at A. Given that AB = 15 cm, find the radius of the circle.

• 8 cm

• 9 cm

• 2 cm

• 16 cm

Multiple Choice

Assume lines that look like tangents are tangents.

• 7.5

• 56.25

• 9.39

• 4.5

Multiple Choice

In the diagram, a tangent and a line drawn to the point of tangency form a right triangle. Solve for "?" (the length of the hypotenuse)

• 15
• 6
• -24
• 14

Multiple Choice

Is line AB tangent to the circle?

• YES
• NO
  NO
• Can't Tell
  Can't Tell

Multiple Choice

Find the radius r

• 2
• 3
• 4
• 5

Multiple Choice

A tangent line intersecting with a radius always creates a ____________ angle

• right
• acute
• obtuse
• straight

Multiple Choice

In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm.Then the radius of the circle is

• 25 cm

• 26 cm

• 24 cm

• 10 cm

Multiple Choice

Determine whether segment MP is tangent to the circle.

• YES

• NO