Goal: to find the length of a line segment using the distance formula.

Method 1 Whenever the segments are horizontal or vertical, the length can be obtained by counting. When we need to find the length (distance) of a segment such as we simply COUNT the distance from point A to point B. (AB = 7) We can use this same counting approach for . (CD = 3) Unfortunately, this counting approach does NOT work for which is a diagonal segment.

Method 2 In the last slide, we found the distance between two numbers on a number line. But what happens if the points are not on a straight line like points A and B above?

Method 2 When working with diagonal segments, the Pythagorean theorem can be used to determine the length.

Method 2 Draw a right triangle using points E and F as corners. When you have a right triangle you can use the Pythagorean theorem to find the distance. c2=a2+b2 (Solve for c) Find the distance between points E and F using the Pythagorean theorem.

Work: The distance between E and F is 5 units. Use this method of working with the Pythagorean Theorem whenever you forget the Distance Formula!

Method 3 When working with diagonal segments, use the Distance Formula to determine the length.   

Now let’s use the Pythagorean theorem to develop the distance formula. The distance between the points E = (x1, y1) and F = (x2, y2) is given by the formula:  Length of Hypotenuse Length of Side 1 (a2) Length of Side 2(b2)

E Distance Formula: F hypotenuse/ distance Length of b: Length of Side 2 a: Length of Side 1 Length of Hypotenuse Length of Side 2(b2) Length of Side 1 (a2)

IMPORTANT! Note: x2 is not the same as x2!!!!!! X2 means x - point number 2 and x2 means the square of x.

Now use the distance formula to find the distance for the first example: Work: Coordinates of E: (1, -1) Coordinates of F: (4, -5)

Distance Formula: It doesn't matter which point you start with.  Just start with the same point for reading both the x and y coordinates The Distance Formula can be used to find the lengths of all forms of line segments:  horizontal, vertical and diagonal.   The advantage of the Distance Formula is that you do not need to draw a picture to find the answer.  All you need to know are the coordinates of the endpoints of the segment.

Use the distance formula to find the distance between the following points. 1. (1,6) and (5,1)

Use the distance formula to find the distance between the following points. 2. (-5,2) and (3, -1)

The expression represents the distance formula The expression represents the distance formula. What are the original points? x2 x1 y2 y1 Coordinates (x2, y2) and (x1, y1) (0, -3) and (11, 2)

Midpoint Formula The point halfway between the endpoints of a line segment is called the midpoint. A midpoint divides a line segment into two equal parts.   In Coordinate Geometry, there are several ways to determine the midpoint of a line segment.   Method 1:If the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.  

Example: Find the midpoints of line segments AB and CD. You are asking yourself “What point is halfway between the two?” Find the midpoints of line segments AB and CD. The length of line segment AB is 8 (by counting).  The midpoint is 4 units from either endpoint.  On the graph, this point is (1,4). The length of line segment CD is 3 (by counting).  The midpoint is 1.5 units from either endpoint.  On the graph, this point is (2,1.5)

However, … If the line segments are diagonal, more thought must be paid to the solution.  When you are finding the coordinates of the midpoint of a segment, you are actually finding the average (mean) of the x-coordinates and the average (mean) of the y-coordinates. This concept of finding the average of the coordinates can be written as a formula:

The Midpoint Formula: The midpoint of a segment with endpoints (x1 , y1) and (x2 , y2) has coordinates:

"The Midpoint Formula" sung to the tune of "The Itsy Bitsy Spider" by Halyna Reynolds Surf City, NJ When finding the midpoint of two points on a graph, Add the two x's and cut their sum in half. Add up the y's and divide 'em by a two, Now write 'em as an ordered pair You've got the middle of the two.

Example: Find the midpoint of line segment AB. A(-3,4) B(2,1)

Find the midpoint of the two points using the Midpoint formula. 1. (-2, 4) and (3, 4) 2. (8, -3) and (5, -4)

Example 3: Southwestern Telephone Company uses GPS to map the locations of its telephone poles. IT is determined that an additional pole is needed exactly halfway between 2 poles located at coordinates (20, 35) and (40, 15). What are the coordinates of the location of the new pole?

4. The expression represents the midpoint formula 4. The expression represents the midpoint formula. What are the original points? x2 x1 y2 y1 Coordinates (x2, y2) and (x1, y1) (-1, 2) and (5, 0)

Trigonometry: For Right Triangles Only! hypotenuse - always opposite the right angle leg leg

Basic Trigonometry Rules: With right triangles you can use three special ratios to solve problems. These ratios ONLY work in a right triangle. The hypotenuse is across from the right angle. Questions usually ask for an answer to the nearest units. You need a scientific or graphing calculator.

Definitions: Hypotenuse Leg a is opposite <A and adjacent to <B. Leg b is opposite <B and adjacent to <A.

Sine Hypotenuse B a C A b c The sine (sin) of an acute angle of a right triangle is the ratio that compares the length of the leg opposite the acute angle to the length of the hypotenuse.

Cosine Hypotenuse B a C A b c The cosine (cos) of an acute angle of a right triangle is the ratio that compares the length of the leg adjacent the acute angle to the length of the hypotenuse.

Tangent Hypotenuse B a C A b c The tangent (tan) of an acute angle of a right triangle is the ratio that compares the length of the leg opposite the acute angle to the length of the leg adjacent the acute angle .

Formulas: Soh Cah Toa Hypotenuse Leg Adjacent to A Leg Opposite A A A represents the angle of reference Leg Adjacent to A Leg Opposite A

Remember:  The formulas can be remembered by: soh cah toa The formulas can be remembered by: oscar had a heap of apples The formulas can be remembered by: oh heck, another hour of algebra!

Examples: Find the sine, cosine and tangent ratios for both acute angles of the following right triangle. C A B 6 10 8

Examples: Find the sine, cosine and tangent ratios for both acute angles of the following right triangle. C A B 6 10 8

Using Trig Ratio Tables/Calculator Find the following values to the nearest ten-thousandth: 0.6691 sin 42° 0.8660 cos 30° 0.5095 tan 27° 0.9563 sin 73° cos 36° 0.8090 tan 81° 6.3138

Applications: In order to determine how an object is grown, you will need to determine the height of the object. You can use trigonometry, or the study of triangles, to find the height of an object. The tangent function can help find the height of objects. To determine the height of the flagpole, set up a triangle with one side being the height of the flagpole (a), another side being a distance from the flagpole to a point on the ground (b), and the third side being the distance from that point to the top of the flagpole (c). Assume the flagpole meets the ground at a right angle.

Solution: The height of the flagpole is 8 meters high. You have two options to solve this problem: 1. Use inverse operations: The height of the flagpole is 8 meters high.

Method 2: Use a proportion. The height of the flagpole is 8 meters high.

Example 2: In right triangle ABC, hypotenuse AB=15 and angle A=35º. Find leg BC to the nearest tenth. Set Up: Draw a picture depicting the situation. Be sure to place the degrees INSIDE the triangle. Place a stick figure at the angle as a point of reference. Thinking of yourself as the stick figure, label the opposite side (the side across from you), the hypotenuse (across from the right angle), and the adjacent side (the leftover side). Notice how the items on the sides of the triangle pair up.  The h pairs with the 15, the o pairs with the x, but the a stands alone.  The a has no companion term.  This means that the a is NOT involved in the solution of this problem.  Cross it out! This problem deals with o and h which means it is using sin A.

Example 3: In right triangle ABC, leg BC=20 and angle B = 41º. Find the hypotenuse BA to the nearest hundredth

Example 4: A ladder 6 feet long leans against a wall and makes an angle of 71º with the ground.  Find to the nearest tenth of a foot how high up the wall the ladder will reach. C A B 6 x

The person in the drawing is using a hypsometer. A hypsometer is an instrument you can use to find the height of very tall objects. How is this example different from the one in our example 1? How will that affect the answer? What will you have to do to compensate for this difference? Will your answer be the same?

Using a Hypsometer: Hold the hypsometer up and look through the straw at the top of a tall object. The imaginary line that goes from your eye to the top of the object is called the line of sight. Once you have found your line of sight, look at the point at which the string crosses the semicircle. That is the measure of the angle from your line of site and an imaginary horizontal line.

Work x Now a = x + 2 A = 10 m Height of person from toes to eye level.

Another Example Now h = x + 2 A = 19.32 m h Height of person from toes to eye level.

Activity: You are now going to use your hypsometer to measure tall objects. Complete the data table on the back To make hypsometer, you will need a straw, some string, a paper clip, and index cared and a protractor (paper). Glue the semi-circle paper protractor) to the index card and cut it out. Attach the straw to the semicircle, with the middle of the straw at the dot labeled A. Attach a piece of string 6 inches long to the center of the straw. Attach a paper clip to the other end of the string. Make sure that the paper clip hangs freely.

Date Table: Length of your foot (in feet): Your height from foot to eye (in feet): Object Angle Ground Distance from Object Tree Building Light Pole Your Choice Now use your data to calculate the height of each object. Show ALL work!