Splash Screen

Concept When the means of a proportion are the same number, that number is called the geometric mean of the extremes. The geometric mean between two numbers is the positive square root of their product.

Example 1 Geometric Mean B) Find the geometric mean between 3 and 12. The means of the proportion are missing. Start filling in: The other two given numbers are the missing parts of the proportion: Cross multiply and solve: TRY SOLVING ON YOUR OWN FIRST:

Concept Altitude of a triangle: The altitude of a triangle is a segment drawn from a vertex to the line containing the opposite side and perpendicular to the line containing that side. Geometric Means in Right Triangles: In a right triangle, the altitude drawn from the vertex of the right angle to the hypotenuse forms two additional right triangles. These three right triangles share a special relationship. This is a right Δ with an altitude drawn to the hypotenuse. This is the figure we will see in the rest of this section.

Example 2 Identify Similar Right Triangles Write a similarity statement identifying the three similar triangles in the figure.

By definition of similar polygons, you can write proportions comparing the side lengths of these triangles. Notice that the circled relationships involve geometric means. This leads to the next theorem. Method 1: use similar Δs from ch 7 to compare corresponding sides. Small Δ: medium Δ Large Δ : small Δ Large Δ : medium Δ WE DO NOT USE THIS METHOD, but still use the geometric mean pattern!!

Concept 1) The altitude is the geometric mean, 2) The short leg is the geometric mean, Or 3) The long leg is the geometric mean When the short or long leg is used for the geometric mean, make sure you use the entire hypotenuse length in one of the two remaining spots of the proportion. Method 2: Set up one of THREE possible proportions

Geometric Mean To help remember the proportion set up, we use this story. Meet Parachute Pete. Pete can parachute down 1 of 3 possible paths.

The paths that Pete follows to the ground are the Geometric Mean. These distances are placed diagonally in the proportion! He can parachute the short leg- He can parachute the altitutde- He can parachute the long leg- Use worksheet “Parachute Pete Story” with the next 3 slides

= == Therefore you can write 3 proportions to find missing side lengths! A C D B BC AD CA BD DC CD BA AD AC So the path he parachutes is the geometric mean. The story continues with him landing on the ground and having one plane ticket to visit another city. If he “parachutes” down a leg, then once he “lands” he can visit the close city or the far city. (WATCH COLOR CODED ANIMATION) “Parachute”, then fill in the geometric mean of the proportion. If he “parachutes” down the altitude, fill in the geometric mean. Then, once he “lands” he can visit the left city or the right city. (WATCH COLOR CODED ANTIMAION) If he “parachutes” down a leg, fill in the geometric mean. Then, once he “lands” he can visit the close city or the far city. (WATCH COLOR CODED ANIMATION) Again, notice how the “far city” uses the entire hypotenuse length when using a leg as the geometric mean.

Parachute Pete, Pg. 36 in WB *Pete always parachutes from the RIGHT ANGLE of the LARGE triangle. *The path he travels is the GEOMETRIC MEAN (so it is used TWICE in the proportion). *Then he visits TWO cities: (these are the other two blanks in the proportion). If the path he traveled was the ALTITUDE (middle) path, then he visits the LEFT city or the RIGHT city. If the path he traveled was an OUTSIDE path, then he visits the CLOSE city or the FAR city. C A DB B C D A Pete will start at C. Pete will start at A.

C B A = PB BA BC Parachute Pete, Pg. 37 in WB For these problems, use the highlighted segment as Pete’s path to parachute.

PA AB = AC PA C B A

C B A = PC CA CB

Set up the proportion for each picture. Use the bold segment as Pete’s Path W O LF D A YS P A T R C A L I Back to the “Parachute Pete Story” worksheet TRY THESE ON YOUR OWN, THEN USE ANIMATION TO GET ANSWERS TO APPEAR

Set up two proportion for each picture. Use SEGMENT NAMES in the first; Numbers and variables in the second. Solve each proportion for x K I TE x 4 9 T U R N x 4 5 C E S H x 9 7 C E S H 8 3 x “Parachute Pete Story” worksheet TRY THESE ON YOUR OWN, THEN USE ANIMATION TO GET ANSWERS TO APPEAR x = 6 x = 5.4 Remember simplifiying radicals? See next slide for review

Simplifying Radicals (Square Roots) - No Decimal Answers when you solve by square rooting (unless there is a decimal under the square root). - Decimals are OK when you solve using division! Pick any 2 numbers that multiply to get Look for pairs of numbers at the bottom of each branch. One number from each pair goes on the outside of the radical. Any numbers without a pair get multiplied on the inside of the radical.

Example 3 Use Geometric Mean with Right Triangles Find c, d, and e. When solving, sometimes it will matter what variable you need to find first. If you set up a proportion and more than one variable is in that proportion, try another path. To solve for c: To solve for d: To solve for e: PETE Back to WB Pg. 37!!!

Example 3 Find e to the nearest tenth. PETE To solve for e:

Example 4 Indirect Measurement KITES Ms. Alspach is constructing a kite for her son. She has to arrange two support rods so that they are perpendicular. The shorter rod is 27 inches long. If she has to place the short rod 7.25 inches from one end of the long rod in order to form two right triangles with the kite fabric, what is the length of the long rod? x The long rod will be x, so Parachute to solve for x. x = = in

Example 4 AIRPLANES A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, at point E. If AE is 163 feet, what is the length of the aircraft to the nearest tenth of a foot? x The length of the plane will be x, so Parachute to solve for x. x = = ft