DO NOW 1215 Triangle ADB is similar to

DO NOW 12/15: Triangle ADB is similar to triangle ABC. Find the value of x. (HINT: Use the similarity statement!) 5 m 10 m x TRIANGLE THEOREMS Agenda 1. Exit Ticket Corrections 2. Triangle Inequality Theorem 3. Hinge Theorem 4. Debrief Learning Target I can explain Triangle Inequality Theorem and Hinge Theorem, and relate it to triangle similarity.

EXIT TICKET

SPAGHETTI TIME! (PART I) Break the stick of spaghetti into ANY three parts. Measure each of the three parts. On your sheet, identify each segment as shortest, middle and longest based on the measures. 1. Did anyone have an equilateral triangle? Isosceles? Scalene? 2. Add the shortest and middle sides together and compare to the length of the longest. Record your observations. If possible, use three segments to make a triangle. Trace the triangle onto a blank/graph piece of paper. 3. Refer back to your answers from #2 above. What was true of the lengths of the 3 sides that could be made into triangles?

TRIANGLE INEQUALITY THEOREM In order to form a triangle, the sum of the two smaller sides of a triangle must be greater than the third side. Ex. AB + BC > AC B A C **MUST BE IN NOTES!**

SPAGHETTI TIME! (PART II) Once you have your triangle traced on your paper, label the shortest side AB, the middle side BC, and the longest side AC. Use a protractor to measure angle A, angle B and angle C. Compare the measures. 1. Which measures are biggest? Which measure is smallest? Which measure is in between? 2. How is this related to the lengths of the sides?

TABLE OF TRIANGLE PARTS Side Angle Smallest Middle Longest What relationship do we notice between the location of sides and angles with relative measures?

HINGE THEOREM The measure of the angles in a triangle are directly related to the lengths of the segments opposite from them. Ex. If AC > BC > AB, then angle B > angle A > angle C A B C **MUST BE IN NOTES!**

DEBRIEF: TRIANGLE THEOREMS How are the sides and angles of triangles related? How can these two theorems help us when we are trying to determine if two triangles are similar?

DO NOW 12/16: Use a similarity statement and proportions to find the length of y. SPECIAL RIGHT TRIANGLES Agenda 1. Embedded Assessment Review 2. Construct/Bise ct Equilateral Triangles (3060 -90) 3. Construct Isosceles Right Triangles (4545 -90) 4. Compare Ratios 5. Debrief Learning Target I can determine the ratios of the sides of all 45 -45 -90 and 30 -60 -90 right triangles.

EMBEDDED ASSESSMENT REVIEW How far will the judge travel from the start to the 2 n d buoy How far will he be at the 2 n d buoy to the finish? How far is the 2 n d buoy from the marker?

CONSTRUCT A SCALENE RIGHT TRIANGLE 1. 2. 3. 4. 5. 6. Draw segment AB of ANY length. Draw a circle with center point A and radius AB. Draw another circle with center point B and radius BA. Label the 2 points of intersection of the two circles C and D. Draw triangle ABC. Bisect the triangle by drawing segment CD. Use the table below to record the side lengths, angle measures and ratios of the two resulting right triangles. Shorter Leg Long/Short Angle A Longer Leg Hyp/Long Angle M Hypotenuse Hyp/Short Angle C

CONSTRUCT AN ISOSCELES RIGHT TRIANGLE 1. Draw segment AB of ANY length. 2. Draw the perpendicular bisector of AB 1. 2. 3. 4. Draw circle A with radius greater than half AB Draw circle B with radius greater than half AB Label the two points of intersection of the circles C and D Connect points CD to bisect AD, marking the point of intersection point M 3. Use the compass to create two congruent segments MA and MB, and connect AB to form a triangle. Use the table below to record the side lengths, angle measures and ratios of the resulting right triangle. Leg/Leg Angle M Leg Hyp/Leg Angle A Hypotenuse Hyp/Leg Angle C

SPECIAL RIGHT TRIANGLES: 30 -60 -90 All right triangles with angle measures 30 -60 -90 will have similar side ratios of 1: 2: √ 3 A 2 x 60° x 30° B C x√ 3 **MUST BE IN NOTES!**

SPECIAL RIGHT TRANGLES: 45 -45 -90 All isosceles right triangles with angle measures of 45 -45 -90 will have side ratios of 1: 1: √ 2 B x x A 45° C x√ 2 **MUST BE IN NOTES!**

DEBRIEF: SPECIAL RIGHT TRIANGLES How are special right triangles related to similarity? How is Hinge Theorem and Triangle Inequality related to special right triangles?

DO NOW 12/17: Below are two special right triangles. Use the ratios to determine the missing sides. TRIG RATIOS Agenda 1. Examples 1 -6 2. Right Triangle Ratios in Groups 3. Examples 7 -10 4. Debrief Learning Target I can identify the opposite, adjacent, and hypotenuse of a right triangle, and use the ratios to determine triangle properties.

LESSON 25: TRIGONOMETRIC RATIOS

LABELING A RIGHT TRIANGLE

GROUP 1

GROUP 2

DEBRIEF/EXIT TICKET Are ratios easier/harder than similar triangles? The same? What about it and why? What are some applications of these ratios in the real world?

DO NOW 12/18: Use the ratios of special right triangles to find the missing variables. SIMPLIFYING RADICALS Agenda 1. Factor Tree/Prime Factorization 2. Adding and subtracting radicals 3. Packet Work 4. Debrief Learning Target I can simplify and perform operations with radicals and use radicals to solve problems with right triangles.

SIMPLIFYING RADICALS A radical is another name for a “root” (like “square root”). To simplify a radical means to reduce the radicand by factoring until there are no more perfect “roots” under the radical sign. Ex. √ 75 **MUST BE IN NOTES!** Ex. √ 32 x 2

OPERATIONS WITH RADICALS The key to operating with radicals is to think of them like variables. You can multiply and divide them together, but you can only add or subtract if they are like terms. Multiplication/Division √ 5 × √ 2 = √ 10 √ 12 ÷ √ 2 = √ 6 Addition/Subtraction 4√ 6 + 3√ 6 = 7√ 6 8√ 3 - √ 75 = 3√ 3 ** No radicals can be in the denominator!** 4 √ 3

PRACTICE WITH SPECIAL RIGHT TRIANGLES AND RADICALS

PROBLEM SET

DEBRIEF/EXIT TICKET When are some common situations when we use radicals in trigonometry? When will simplifying radicals be better than using a decimal?

DO NOW 12/19: Eat a candy cane and take your quiz. SPECIAL RIGHT TRIANGLES QUIZ Agenda 1. Eat Candy Canes 2. Take a quiz 3. Finish your packet/Do extra credit work 4. Debrief

RIGHT TRIANGLES WITH AN ALTITUDE OR “GEOMETRIC MEAN”

SPECIAL RIGHT TRIANGLES

DEBRIEF Have a safe and relaxing holiday!