4_SpecialRightTriangles.pptx triangles special

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1. Special Right Triangles

2. Warm Up Use the Pythagorean Theorem to find the unknown lengths to the nearest tenth. 1. 2.

3. Warm Up Use the Pythagorean Theorem to find the unknown lengths to the nearest tenth. 3. 4.

4. Warm Up 5. MONITORS Maria thinks that the largest monitor her small desk will hold is 10 inches tall by 16 inches wide. On an online shopping site, she finds a 22-inch monitor with the same aspect ratio, measured diagonally. Will the computer fit on Maria’s desk?

5. Warm Up Use the Pythagorean Theorem to find the unknown lengths to the nearest tenth. 1. 13 2. 8

6. Warm Up Use the Pythagorean Theorem to find the unknown lengths to the nearest tenth. 3. 8.1 4. 9.5

7. Warm Up 5. MONITORS Maria thinks that the largest monitor her small desk will hold is 10 inches tall by 16 inches wide. On an online shopping site, she finds a 22-inch monitor with the same aspect ratio, measured diagonally. Will the computer fit on Maria’s desk? no

8. Standards for Mathematical Content G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

9. MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP8 Look for and express regularity in repeated reasoning. Standards for Mathematical Practice

10. Lesson Goals ● Understand that by similarity, side ratios in 45-45-90 right triangles are related to the angles in the triangles. ● Understand that by similarity, side ratios in 30-60-90 right triangles are related to the angles in the triangles.

11. Learn Triangles   The diagonal of a square forms two congruent isosceles right triangles. Because the base angles of an isosceles triangle are congruent, the measure of each acute angle is or 45°. Such a special right triangle is known as a triangle.  

12. Learn Triangles   Theorem 9.8: Triangle Theorem In a triangle, the legs 𝓁 are congruent and the length of the hypotenuse h is times the length of a leg.

13. Learn Triangles   Talk About It! Why do all triangles have the same side length ratios? Use similarity to justify your reasoning.  

14. Example 1 Find the Hypotenuse Length Given an Angle Measure Find the value of .  

15. Example 1 Find the Hypotenuse Length Given an Angle Measure The acute angles of a right triangle are complementary, so the measure of the third angle is or Because this is a triangle, use the Triangle Theorem.   Triangle Theorem Substitution Substitution

16. Example 1 Find the Hypotenuse Length Given an Angle Measure Think About It! How can you remember the ratios of the side lengths of a triangle?  

17. Example 1 Find the Hypotenuse Length Given an Angle Measure Check Find the value of .  

18. Example 1 Find the Hypotenuse Length Given an Angle Measure 14 Check Find the value of .  

19. Example 2 Find the Hypotenuse Length Given a Side Measure Find the value of .  

20. Example 2 Find the Hypotenuse Length Given a Side Measure The legs of this right triangle have the same measure, so the triangle is isosceles. Because this is a triangle, use the Triangle Theorem.   Triangle Theorem Substitution or Solve. Substitution Solve.

21. Example 2 Find the Hypotenuse Length Given a Side Measure Check Find the value of .  

22. Example 2 Find the Hypotenuse Length Given a Side Measure   Check Find the value of .  

23. Example 3 Find Leg Lengths in a Triangle   Find the value of  

24. Example 3 Find Leg Lengths in a Triangle   The acute angles of a right triangle are complementary, so the measure of the third angle is or 45°. So, the triangle is a triangle. Use the Triangle Theorem to find the value of .  

25. Example 3 Find Leg Lengths in a Triangle   Triangle Theorem Substitution Divide each side by . Substitution The value of is or .  

26. Example 3 Find Leg Lengths in a Triangle   Check Find the value of .  

27. Example 3 Find Leg Lengths in a Triangle   Check Find the value of    

28. Learn Triangles   A triangle is a special right triangle or right triangle with side lengths that share a special relationship. You can use an equilateral triangle to find this relationship. When an altitude is drawn from any vertex of an equilateral triangle, two congruent triangles are formed. In the figure, so . If , then and . Because is equilateral, and .  

29. Learn Triangles   Use the Pythagorean Theorem to find , the length of the altitude , which is also the longer leg of .   Pythagorean Theorem Simplify. Subtract from each side. Simplify. Pythagorean Theorem Simplify. Simplify.

30. Learn Triangles   Theorem 9.9: Triangle Theorem In a triangle, the length of the hypotenuse is times the length of the shorter leg , and the longer leg 𝓁 is times the length of the shorter leg.

31. Learn Triangles   Think About It! Ian states that in a triangle, sometimes the angle is opposite the longer leg and the angle is opposite the shorter leg. Do you agree or disagree with Ian? Justify your answer.  

32. Example 4 Find Leg Lengths in a Triangle   Find the values of and .  

33. Example 4 Find Leg Lengths in a Triangle   Use the Triangle Theorem to find the value of , the length of the shorter side.   Triangle Theorem Substitution or Divide each side by . Substitution

34. Example 4 Find Leg Lengths in a Triangle   Now use the Triangle Theorem to find the value of , the length of the hypotenuse.   Triangle Theorem Substitution or Simplify. Substitution Simplify.

35. Example 4 Find Leg Lengths in a Triangle   Check Find the values of and .      

36. Example 4 Find Leg Lengths in a Triangle   Check Find the values of and .   or   or  

37. Apply Example 5 Use Properties of Triangles   JEWELRY Destiny makes and sells upcycled earrings. The earrings shown are made from congruent equilateral triangles. Each triangle has a height of centimeters. The hooks attached to the top of the earrings are centimeter tall. Destiny needs to mail this pair of earrings to a customer. If she mails the earrings in a rectangular box, what width and length must the base of the box have so the earrings will fit if they are placed side by side in the bottom of the box?  

38. Apply Example 5 Use Properties of Triangles   Think About It! What assumption did you make while solving this problem?

39. 2. The perimeter of an equilateral triangle is . What is the length of the altitude?   1. The perimeter of a square is . What is the length of the diagonal?   Exit Ticket 3. Find the missing side lengths on the figure.

40. 2. The perimeter of an equilateral triangle is . What is the length of the altitude?   1. The perimeter of a square is . What is the length of the diagonal?   Exit Ticket 3. Find the missing side lengths on the figure. , , , ,