Warm Up Find x. Leave answer as simplified radical

Special Right Triangles 30 – 60 – 90

Find the missing side lengths 60º 30º 10 Can we use Pythagorean Theorem? Can we use similar Triangles? Any other way we know of to find the missing side lengths? This is the intro. Use this slide to get the students talking about previous knowledge (pythag. Theorem and similar triangles) AND to introduce the idea of finding side lengths using special angles. Unless you have a super star in your class, the students will not be able to answer this question so use it to start the guided task.

Before we begin… ALTITUDE HYPOTENUSE LEG Lets go over some vocabulary needed ALTITUDE The perpendicular height from one side of a triangle to the opposite vertex HYPOTENUSE The longest side of a right triangle (the side across from the right angle) LEG The two sides that connect to the right angle in a right triangle. If you have been keeping a notebook or word wall of vocabulary this would be a good place to add these words. These should be words they have already learned but may not remember. If your students already remember/know these words you can move on.

Discovering Special Triangles 1. Adam, a construction manager in a nearby town, needs to check the uniformity of Yield signs around the state and is checking the heights (altitudes) of the Yield signs in your locale. Adam knows that all yield signs have the shape of an equilateral triangle. Why is it sufficient for him to check just the heights (altitudes) of the signs to verify uniformity? You may choose to differentiate here depending on the ability level of your students. Higher students should be able to complete this task with only minor facilitation where lower students will need a more guided learning experience. I have including the guided experience here in the PPT where there are many pauses and checks for understanding before moving along. Also the concepts of WHY are explained more than discovered. GUIDED: In Pairs have students discuss this for a few minutes then come back as a full group and answer this question. Because all equilateral triangles are similar so one measurement will be sufficient.

Discovering Special Triangles 2. A Yield sign from a street near your home is pictured to the right. It has the shape of an equilateral triangle with a side length of 2 feet. If the altitude of the triangular sign is drawn, you split the Yield sign in half vertically, creating two 30°-60°-90° right triangles, as shown to the right. For now, we’ll focus on the right triangle on the right side. (We could just as easily focus on the right triangle on the left; we just need to pick one.) We know that the hypotenuse is 2 ft., that information is given to us. The shorter leg has length 1 ft. Why? Again, this slide is meant for explanation for lower students. They will find it difficult remembering Triangle congruence Use next slide for explanation if needed

2 Congruent due to HL 2 So the two bottom legs must be congruent 2 We know all sides have a length of 2. So if that side is split into 2 congruent pieces each piece must be 1. 1

3. What is the length of the third side (the altitude) 3.What is the length of the third side (the altitude)? Leave answer as simplified radical. X 2 1 Pythagorean Theorem: 12 + x2 = 22 1 + x2 = 4 x2 = 3 x =

Quick Review: How do we simplify radicals? Break down the radicand (the number inside the radical) into perfect squares. Anything that is a perfect square will come out of the radical everything else stays inside the radical. 12 10 Only use this slide if you need review of simplifying radicals 5 2 6 3 2

Quick Review: How do you rationalize the denominator? We can never leave a radical in the denominator. Multiply the numerator and denominator by the radical on the bottom. This will get rid of the radical on the denominator, then simplify. 18 3 ∙ 3 3 = 18 3 3 = 6 3 Only use this slide if you need review of rationalizing the denominator

Answer question 4 on your own or in your pair NOTE!! When simplifying for the y value students will have to simplify .75 which can get very weird if they have never simplified decimals/fractions before.

5) Now that we have found the altitudes of both equilateral triangles, we look for patterns in the data. Fill in the first two rows of the chart below, and write down any observations you make. Then fill in the third and fourth rows. Side Length of Equilateral Triangle Each 30°- 60°- 90° right triangle formed by drawing altitude Hypotenuse Length Shorter Leg Length Longer Leg Length 2 (first)   1 (second) 4 6 This question is great for differentiating. Higher students can solve 5 and 6 on their own, but for lower students help facilitate them to the answer. Do this by filling out the chart and asking questions like, “Rather than drawing a triangle for side length of 4 do you notice a pattern?”

5) Now that we have found the altitudes of both equilateral triangles, we look for patterns in the data. Fill in the first two rows of the chart below, and write down any observations you make. Then fill in the third and fourth rows. Side Length of Equilateral Triangle Each 30°- 60°- 90° right triangle formed by drawing altitude Hypotenuse Length Shorter Leg Length Longer Leg Length 2 (first)  2 1    1 (second)  1  1/2 4 4  2  6 6   3 This question is great for differentiating. Higher students can solve 5 and 6 on their own, but for lower students help facilitate them to the answer. Do this by filling out the chart and asking questions like, “Rather than drawing a triangle for side length of 4 do you notice a pattern?”

6. What is true about the lengths of the sides of any 30°-60°-90° right triangle? 2x x x 3 30º 60º Give students 1-2 minutes to discuss and come up with a hypothesis and then have them share aloud. Answers will be slightly different depending on how they determine/see the patterns.

Foldable! Use the instructions on how to make the foldable. Feel free to change it in any way that will most benefit your students. I use the examples that we write in there as guided notes. That way they are not taking down notes and THEN making a foldable. They are doing it all at once.

Once you have made your foldable complete the table for question 7