Warm-Up: April 10, 2015. Homework Questions? Geometry in the Coordinate Plane Investigation 8C Advanced Integrated Math I.

Warm-Up: April 10, 20151. On graph paper, draw a rectangle with vertices at , , and .2. Find the coordinates of the points described below.

a) The fourth vertexb) Four points that are inside the rectanglec) Four points that are outside the rectangled) Four more points that lie on the rectangle

3. How can you tell whether a point is inside the rectangle just by looking at its coordinates?

Homework Questions?

Geometry in theCoordinate Plane

Investigation 8CAdvanced Integrated Math I

You-Try #2

Suppose you have points and .a) What is the distance between and ?b) Find the coordinates of the midpoint of

You-Try #3a) How many vertical lines contain the point ?b) Name the coordinates of the intersection of a horizontal line through

and a vertical line through .c) Plot the points , , and . Find the lengths of all three sides of

You-Try #4Suppose that and .a) Find the coordinates of the midpoint of .b) Find the length of .

Assignment

• Page 660 #5, 7, 8

Page 660

See textbook for #7

Warm-Up: April 13, 2015

• Copy and complete the table below.

Homework Questions?

Midpoint and Distance Formulas

Section 8.11Advanced Integrated Math I

You-Try #1

• Here is a claim about the vertices of square :

a) Is the claim true when ? That is, is it true that ?b) Is the claim true when ?c) Is the claim true when ?d) Is the claim true when ?

You-Try #2

• Here is another claim about the vertices of square :If has coordinates , then has coordinates .

a) When , the claim says “If has coordinates , then has coordinates .” Look at the table and decide whether this is true.

b) Find a value of for which the statement does not make sense.

Minds in Action

• Sasha and Derman are trying to write a formula for finding the distance between two points given by and .• Sasha: Well, the distance between two points on a vertical or

horizontal line is easy to find. Just subtract the unlike coordinates. If is and is , then the distance between and is .• Derman: But we don’t know that and are on a horizontal or vertical

line. They’re just any two points. We used the Pythagorean Theorem to help us find the distance between two points that weren’t on the same horizontal or vertical line before. Let’s try that here.

Minds in Action

• Sasha: Don’t we need three points to make a triangle so we can use the Pythagorean Theorem?• Derman: Watch! I’ll make a third point.

• Derman: Before you ask, I know that the third point is . In my picture I had to go over as far as the point – that’s where I got the ; and up as far as the point – that’s where I got the

Minds in Action

• Sasha: Great! Now let’s use the Pythagorean Theorem to find the length of the hypotenuse of that triangle

So,

Distance Formula

• The distance between two points and can be found using the Pythagorean Theorem:

You-Trys

• Find the distance between each pair of points.7. and 8. and 9. and

Midpoint Formula

• The midpoint of a line segment with endpoints and can be found by averaging the -coordinates and averaging the -coordinates:

You-Trys

10. Find the midpoint of the segment with endpoints and .

11. Find the midpoint of the segment with endpoints and .

Assignment

• Read Section 8.11 (pages 662-665)• Page 666 #4, 8, 10, 11, 12

Page 6664. Write about it: Explain how to tell whether two triangles are

congruent by doing calculations on the coordinates of their vertices.8. Three vertices of a square are , and .

a. Find the center of the square.b. Find the fourth vertex.

Warm-Up: April 14, 2015Create six different graphs according to the following directions:1. Plot several points with -coordinates that have the given property. For

each property, draw a picture that shows all the points with that property.a. 1 more than the -coordinateb. 2 more than the -coordinatec. 1 less than the -coordinate

2. Plot several points with -coordinates that have the given property. For each property, draw a picture that shows all the points with that property.a. Twice the -coordinateb. Three times the -coordinatec. Four times the -coordinate

Homework Questions

Parallel Lines and Collinear Points

Section 8.12Advanced Integrated Math 1

You-Try (with partner)

3. Some of the lines you drew in the warm-up have points with coordinates of the form .a. What do those lines have in common?b. Write the equations of each of those lines in the form .c. What do these equations have in common?

4. Some other lines you drew in the warmup have points with coordinates of the form .a. What do those lines have in common?b. Write the equations of each of those lines in the form .c. What do these equations have in common?

Theorem 8.4

• Two nonvertical lines are parallel iff they have the same slope.

• The proof of Theorem 8.4 depends on Postulate 8.2:• Points , , and are collinear iff the slope between and is the same as

the slope between and ., , and are collinear

Think-Pair Share

• What is true about points , , and and the lengths of segments when moves onto ?

Example

• Decide whether the following three points are collinear:

You-Trys

• Decide whether each set of three points are collinear.

Assignment

• Read Section 8.12 (pages 669-671)• Page 672 #5-9

Page 672

Warm-Up: April 15, 2015

1. Find and .2. What is the slope of ?3. What is the slope of ?4. What is the product of

the two slopes you found in #2 and #3?

Homework Questions?

Perpendicular LinesSection 8.13

Advanced Integrated Math I

Perpendicular Lines

• Theorem 8.5: Two nonvertical lines are perpendicular iff the product of their slopes is .

• Any horizontal line is perpendicular to any vertical line.

Minds in Action

• Tony wonders whether he has just seen a special case.• Tony: Hey Sasha, suppose the lines don’t intersect at the origin. Is

Theorem 8.5 still true?• Sasha: Well, Theorem 8.5 doesn’t say anything about the lines having

to intersect at the origin. But I agree that we have proved one direction of the theorem only for lines like that. Let’s try to prove it for lines that could intersect anywhere.• Tony: Let’s try to avoid a lot of work. Why don’t we just try

translating?

Minds in Action

• Sasha: That should work. The proof we came up with in the warm-up didn’t really use the fact that the lines were through the origin. Look at this diagram:

Minds in Action• Tony: Wow, that’s true! Everything is the same as before:

and

Knowing that and are perpendicular and that and are right angles, we know that and are similar, So

This means that

Minds in Action

• Sasha: Good, Tony! You really got it! So what are the slopes of and ?• Tony: That’s easy! The slope of is , just as before. The slope of is .

Therefore, the product of the slopes is

You-Try #9

• Find an equation of the line through that is perpendicular to line with equation .

You-Try #10

• Find the point of intersection, , of line and the line you just found.

You-Try #11

• What is the distance from to ? (Hint: this is the distance from to .)

Assignments

• Read Section 8.13 (pages 674-676)• Page 677 #5-9

• Page 679 #1-6 (Investigation 8C Reflection)

• Page 680 #1-13 (Chapter 8 Review)

Page 677