Geometry / Mr. Graham, Mr. Kelley, Mr. Hansen

Name: _____________________

Honor signature: _______________

Geometer's Sketchpad Lab

Intersecting Secants, Chords, and Tangents

IMPORTANT / PLEASE READ: I pledge that the words on this assignment are my own. Although working with a partner is permitted, I understand that the recording of data and the writing of conjectures are activities I must do on my own, without copying. By my signature above and my initials here (______), I hereby agree to these terms.

Part I.

Open a new sketch. Using the circle tool, place a circle on the sketch.

Select the circle, and use CONSTRUCT POINT ON OBJECT three times so that there are 4 points on your circle.

Label the points on the circle A , B, D, and C (in that order, traveling clockwise). Label the center of the circle O.

Construct the lines AD and BC.

Use CONSTRUCT POINT AT INTERSECTION, and label that point X.

Move points, if necessary, so that X is in the interior of the circle, but not at the center.

Measure the lengths of segments AX, XD, BX, and CX (see below for space to record answers).

Select these measures and use CALCULATE to find these two products:

1) (AX)(XD)

2) (BX)(XC)

Write your results here:

AX = ____________

XD = ____________

BX = ____________

CX = ____________

(AX)(XD) = ____________

(BX)(XC) = ____________

Now, move points so that X is outside the circle, with DX < AX.

Look at the new values of your measurements. Write your new values here:

AX = ____________

XD = ____________

BX = ____________

CX = ____________

(AX)(XD) = ____________

(BX)(XC) = ____________

Now, move points so that X is outside the circle, with DX > AX.

Look at the new values of your measurements. Write your new values here:

AX = ____________

XD = ____________

BX = ____________

CX = ____________

(AX)(XD) = ____________

(BX)(XC) = ____________

Now, move points so that points A and D coincide. DX (or AX) is now tangent to the circle.

Look at the new values of your measurements. Write your new values here:

AX = ____________

XD = ____________

BX = ____________

CX = ____________

(AX)(XD) = ____________

(BX)(XC) = ____________

Write your observations in the form of a conjecture (or conjectures).

 

 

 

Solve these problems:

Problem 1:

If AX = 8, XD = 3 and CX = 4, show work and find BX. Answer: BX = __________ .

 

Problem 2:

If AD = 3, DX = 3 and BX = 2, show work and find BC. Answer: BC = __________ .

 

Part II.

Delete all measurements from your sketch. Move points so that A, B, C and D are distinct points and X is in the interior of your circle, and construct line segments AB, CD, AC, BD.

You should be able to find two different pairs of similar triangles.

D ______ ~ D______ and D_____~ D______ (You may use Sketchpad measurements of angles or segments to help you decide which triangles are similar.)

Now, move the points so that X is outside the circle, with DX < AX.

Do not construct or label the point of intersection of AB and CD.

Using only labeled points, you should be able to find two different pairs of similar triangles.

D ______ ~ D______ and D_____~ D______ (You may use Sketchpad measurements of angles or segments to help you decide which triangles are similar.)

Now, move the points so that X is outside the circle, with DX > AX.

Do not construct or label the point of intersection of AB and CD.

Using only labeled points, you should be able to find two different pairs of similar triangles.

D ______ ~ D______ and D_____~ D______ (You may use Sketchpad measurements of angles or segments to help you decide which triangles are similar.)

Now, move the points so that X is outside the circle, and points D and A coincide. You should be able to find a pair of similar triangles.

D ______ ~ D______

(You may use Sketchpad measurements of angles or segments to help you decide which triangles are similar.)