Lesson 3

Tangent Lines

  • Let’s explore lines that intersect a circle in exactly 1 point.

Problem 1

Line \(BD\) is tangent to a circle with diameter \(AB\). Explain why the measure of angle \(BCA\) must equal the measure of angle \(ABD\).

Circle. Diameter A B drawn. Point C lies on circle, creating triangle A B C. Point D lies above the circle, between points C and B. Segment A D and D B drawn.   

Problem 2

Line \(AC\) is perpendicular to the circle centered at \(O\) with radius 1 unit. The length of \(AC\) is 1.5 units. Find the length of segment \(AB\).

Circle center O with points B and C on circle. Line A C tangent to circle, segment labeled 1 point 5. Segment A O drawn, passing through B. Triangle A O C drawn.  

Problem 3

Technology required. Line \(PD\) is tangent to a circle of radius 1 inch centered at \(O\). The length of \(PD\) is 1.2 inches. The length of \(AB\) is 1.7 inches. Which point on the circle is closest to point \(P\)?

Circle with points and lines.
A:

point \(A\)

B:

point \(B\)

C:

point \(C\)

D:

point \(D\)

Problem 4

The arc from \(A\) to \(B\) not passing through \(C\) measures 50 degrees. Select all the true statements.

Circle, center O. Points A,  B and C lie on circle. Angles A O B and A C B.  
A:

Angle \(BCA\) measures 50 degrees.

B:

Angle \(BCA\) measures 25 degrees.

C:

Angle \(BOA\) measures 50 degrees.

D:

The arc from \(B\) to \(C\) not passing through \(A\) measures 180 degrees.

E:

Angles \(CBO\) and \(CAO\) are congruent.

(From Unit 7, Lesson 2.)

Problem 5

Chords \(AC\) and \(DB\) intersect at point \(E\). List 3 pairs of angles that must be congruent.

Circle. Points A, B, C and D lie on cirlce. Chords A C and B D intersect at point E. Angles A D B, B C A, C E D and D E A.  
(From Unit 7, Lesson 2.)

Problem 6

The image shows a circle with diameters \(AC\) and \(BD\). Prove that chords \(BC\) and \(AD\) (not drawn) are congruent.

Circle with center E. Diameters A C and B D are drawn.
(From Unit 7, Lesson 1.)

Problem 7

The line represented by \(y+3=\text-3(x+6)\) is transformed by the rule \((x,y)\rightarrow (\text-x,\text-y)\). What is the slope of the image?

A:

3

B:

\(\frac13\)

C:

\(\text-\frac{1}3\)

D:

-3

(From Unit 6, Lesson 12.)