Each student in your group should have a different equilateral triangle. Complete the following steps: Using the centimeter grid paper, determine the length of the side of the triangle. Write the measure on the triangle. Using a compass, set the compass points equal to the length of the side of the equilateral triangle. Draw a quarter of a circle on the grid paper with the center at the origin and the radius equal to the length of a side of the equilateral triangle. Fold the triangle in half in order to form a right triangle. Write the measure of each angle and the length of the short leg of the triangle on this right triangle. Calculate the length of the remaining side of the right triangle. The Pythagorean Theorem will be needed. Leave the answer in simplified radical form. Write the measure on the triangle. Place the vertex of the 30⁰ angle at the origin and the longer leg along x-axis. Write the ordered pair at the point where the triangle intersects the circle. o In a right triangle, the ratio of the length of the leg opposite an angle divided by the length of the hypotenuse is called the sine of an angle. What is the ratio of sin 30⁰? (The abbreviation of sine is sin.) o In a right triangle, the ratio of the length of the leg adjacent to an angle divided by the length of the hypotenuse is called the cosine of an angle. What is the ratio of cos 30⁰? (The abbreviation of cosine is cos.) o In a right triangle, the ratio of the length of the leg opposite of an angle divided by the length of the leg adjacent to an angle is called the tangent of an angle. What is the ratio of tan 30⁰? (The abbreviation of tangent is tan.) Page 1 of 2
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Answer Key Each student in your group should have a different equilateral triangle. Complete the following steps: Using the centimeter grid paper, determine the length of the side of the triangle. Write the measure on the triangle. Using a compass, set the compass points equal to the length of the side of the equilateral triangle. Draw a quarter of a circle on the grid paper with the center at the origin and the radius equal to the length of a side of the equilateral triangle. Fold the triangle in half in order to form a right triangle. Write the measure of each angle and the length of the short leg of the triangle on this right triangle. The angle measures are 30⁰, 60⁰ and 90⁰. Calculate the length of the remaining side of the right triangle. The Pythagorean Theorem will be needed. Leave the answer in simplified radical form. Write the measure on the triangle. Answers will vary depending on the equilateral used. For example, triangle #1 has side lengths of 10 centimeters, 5 centimeters and 5 3 centimeters. Leaving answers in radical form allows students to see patterns. Place the vertex of the 30⁰ angle at the origin and the longer leg along x-axis. Write the ordered pair at the point where the triangle intersects the circle. o In a right triangle, the ratio of the length of the leg opposite an angle divided by the length of the hypotenuse is called the sine Page 1 of 3
sine is sin.) sin 30⁰ = ଵ. In a right triangle, the ratio of the ଶ length of the leg adjacent to an angle divided by the length of the hypotenuse is called the cosine of an angle. What is the ratio of cos 30⁰? (The abbreviation of cosine is cos.) cos 30⁰ = ଷ ଶ o In a right triangle, the ratio of the length of the leg opposite of an angle divided by the length of the leg adjacent to an angle is called the tangent of an angle. What is the ratio of tan 30⁰? (The abbreviation of tangent is tan.) 1 2 1 3 ݎ 3 3 3 2 Page 2 of 3
12 The coordinates are: Aሺ2 3, 2ሻ 10 Bሺ3 3, 3ሻ Cሺ4 3, 4ሻ 8 Dሺ5 3, 5ሻ Eሺ6 3, 6ሻ 6 E D 4 C B 2 A 5 10 15 20 25-2 Page 3 of 3
Scaffolding Notes: If students have not been working with radicals previous to this lesson, a review should be provided including information about when and why we simplify radicals. Assist students struggling with using a compass. What are the measures of each angle of an equilateral triangle? After folding the triangle, which angle did not change? Label that angle. One angle was folded exactly in half. Knowing this, what is the measure of the new angle? Label it. How many degrees are there in a triangle? Knowing this, what is the measure of the third angle? Mark the point where the triangle intersects the circle. What is the x coordinate of this point? What is the y coordinate of this point? Find and label the hypotenuse of the triangle hypotenuse. Using the 30 angle find and label the side opposite the 30 angle as opposite ; label the remaining side as adjacent. Looking at the labels of each of the sides in relation to the 30 angle, why is one side labeled adjacent?