unit 3 lesson 1 inv 1 answers.notebook 3 inv 1 ans.pdfoct 18, 2013 · unit 3 lesson 1 inv 1...

Unit 3 Lesson 1 Inv 1 answers.notebook October 18, 2013


1. In your groups, match the icons below to the options in the Draw menu.

2. Draw a circle that is congruent to the circle shown in the TATS. Explain how you knew it was congruent to the circle in the activity.

What do you believe makes something congruent?

3. Draw a quadrilateral that is congruent to the quadrilateral shown on the TATS. Explain how you knew it was congruent to the quadrilateral in the TATS.

What makes something a quadrilateral?

We will now begin discussing coordinates.

4. What is a coordinate? How do you represent coordinates on a graph?

5. What are vertices? How do you locate vertices on a graph?


6. You will now explore how to use interactive software to create special quadrilaterals. First, clear the window.

a. Draw a rectangle. Record the coordinates of the vertices in the order in which you drew them.

b. How do you know the figure you drew above is a rectangle?

c. Clear the window and find a different way of drawing the same rectangle as you did in part a. Describe the method you used.

d. Clear the window and draw a parallelogram that is not a rectangle. Draw a rough sketch of the parallelogram below.

e. Record the coordinates of the vertices in the order in which you drew them.

f. How did you know the figure you drew in part d was a parallelogram? How was it different from the rectangle you drew before?

g. By clicking and dragging a point, you can generate shapes fro which some conditions remain the same and other conditions vary. Click and drag one of the vertices of your parallelogram in part c. What types of shapes can you create?

horizontal lines are perpendicular to vertical lines...all angles are right angles.

make sure the width and length are the same for both rectangles.

We used slopes to determine that the opposite sides were parallel.


1. On the grid below, draw a rectangle ABCD with coordinates A (2, 9), B (10, 9), C (1, 6) and D (2, 6).

2. Draw the diagonals of the rectangle.

3. Fill in the chart to the right.

Segment slope

AC

BD

Segment Slope

What similarities or differences do you notice in the slopes of the segments?

4. Calculate the slope of the diagonals and fill in the chart below.

Describe the process you used to calculate the slopes of the diagonals. List any formulas you used.

Investigation 1: Calculating Slopes, Lengths, and Midpoints.


Segment Length

5. Fill in the chart below. Determine the length without using a ruler.

What similarities or differences do you notice in the lengths of the segments?

6. Suppose you have two points with coordinates (a, b) and (c, b) with a < c.

a. How do you know that the points are on a horizontal line?

b. Write an algebraic expression for the length of the segment or distance between the points.

7. Suppose you have a vertical line.

a. What is true of the coordinates of all points on a line?

b. Using variables, write coordinates for a point on the same vertical line as the point with coordinates (a, b).

c. Write an algebraic expression for the length of the segment (or distance of the points).


8. Find the lengths of the diagonals in the chart below. Think of a way to do this without a ruler.

9. Now consider points P (1, 3) and Q (2, 7) in a coordinate plane.

Make a sketch on a coordinate grid showing the points P and Q and PQ.

b. Find the length of PQ or distance between points P and Q. compare your answer and method with those of your classmates. Resolve any differences.

c. Use similar reasoning to find the distance between points S (5, 4) and T (3, 2).

Describe the process you used.

Segment Length

AC

BD


10. To generalize the method you used for calculating distance between two points in a coordinate plane, consider general points A (a, b) and B (c, d) graphed below.

a. Make a copy of the diagram showing the coordinates of point C.

b. Write expressions for the distances AC and BC.

c. Write a formula for calculating the distance AB. Compare your formula with that of your classmates and resolve any differences.

d. When points A and B are on a horizontal or vertical line, will your formula calculate the correct distance AB? Why?


Midpoints:1. Draw a rectangle ABCD with the following vertices A (2, 9), B (10, 9), C (10, 6) and

D (2, 6) using the computer.

a. Use the Midpoint command in the Construct menu to find the midpoint of each side of the rectangle.

Record the coordinates of the midpoint: ______________

b. What should the software report as the midpoint of the segment with endpoints P (3, 2) and Q (1, 2)? With endpoints S (3, 3) and T (3, 2)? Check your thoughts.

c. Suppose you have two points P (a, b) and Q (c, b). Write expressions for the coordinates of the midpoint of PQ. Compare your expressions with others and resolve any differences.

d. Repeat part c for the case of points P (a, b) and Q (a, c).

midptAD = (2, 1.5) midptAB = (6, 9) midptBC = (10, 1.5) midptCD = (6, ‐6)

P(‐3, 2) Q(1, 2)

Midpt PQ (‐1, 2)S (3,3) T (3, ‐2)midptST= (3, 0.5)


2. Use the Midpoint command to find the midpoint of diagonal AC of the rectangle in problem 1

Repeat for diagonal BD. Record the coordinates.

a. What do you notice about the midpoints of the diagonals of the rectangle?

b. Now consider a segment with general endpoints A (a, b) and B (c, d). Make a conjecture about the coordinates of the midpoint M of this segment.

i. Test your conjecture using the coordinate grid and the points V (2, 4) and W (6, 8).

ii. Check that your calculated point is the midpoint by verifying that

1. It is on the line containing the points V and W (slope), and

2. It is equidistant from points V and W. (distance formula).

unit 3 lesson 1 inv 1 answers.notebook 3 inv 1 ans.pdfoct 18, 2013 · unit 3 lesson 1 inv 1...

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