Inequalities in Triangles Geometry 5-5

Transcription

1 Inequalities in Triangles Geometry 5-5 Name: ate: Period: Theorem 5-10 Theorem 5-11 If two sides of a triangle are not If two angles of a triangle are not congruent, then the larger angle congruent, then the longer side lies opposite the longer side. lies opposite the larger angle. If XZ > XY, then m Y > m Z. If m > m, then >. 7. raw PQT with m P = 0 and m T = 70. ind m Q. List the sides in of the triangle in order from smallest to largest. 8. raw GHK, where m H = 90, GH =, and HK = 8. ind GK. List the angles in of the triangle in order from smallest to largest. Use the figure at the right for questions Name the longest segment in Δ. 10. Name the shortest segment in Δ Name the shortest segment in Δ. 12. Name the longest segment in Δ ind the shortest segment in the entire figure. 14. How many of the segments in the figure are longer than? 15. List the angles in order from least to greatest.

2 TIVITY: ut 5 strips of paper of lengths: in, 4 in, in, 7 in, and 8 in. Use these strips to complete the chart. o a, b, and c a b c a + b = Is a+b>c? a + c = Is a+c>b? b + c = Is b+c>a? form a triangle? in 4 in in in 4 in 7 in in 4 in 8 in Which set of strips will form a triangle? In a triangle, what is the relationship between the sum of any two sides and the length of the third side? What is another set of strips that will form a triangle? If a triangle has sides of inches and 7 inches, what are the possible lengths of the third side? Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In other words, (a + b) > c. heck that the SUM of the smaller two sides is GRETER THN the largest side. When given two sides of a triangle, finding the possible lengths of the third side () is easy: a + b = maimum a b = minimum minimum < < maimum (Must be positive!) Is it possible to have a triangle with these side lengths? Write yes or no and eplain why , 11, , 7, , 12, , 20, 10 The measures of two sides of a triangle are given. What inequality describes the range of lengths that are possible for the third side? and and and and

3 Review: Simplifying Radicals ind the largest factor that is a perfect square. Eample: 7 = 28 Name: ate: Period: 2 = = Simplify. Show your work. 1.) ( 2 )( 8) 17.) ( )( 2) 18.) ( 5 ) 2 19.) ( )( 2 ) 20.) ( 7 ) 2 21.) ( 2 2) 2

4 Rationalizing the enominator Geometry tetbook pg 755 Name: ate: Period: Generally, it is mathematically improper to have a fraction with a radical in the denominator. To fi this, we perform a procedure called rationalizing the denominator. We want to leave the problem in SIMPLEST ORM: 1. No perfect square factor other than 1 is under the radical sign. 2. No fraction under the radical sign.. No fraction has a radical in the denominator. Multiply the numerator & denominator by the unwanted radical. Eample: = = = = Solve for = 18. = = =

5 What s so special about these Special Right Triangles? Name ate Pd 1. Label the missing angle measures for all angles in this square, and mark any congruent sides with congruency marks. 2. ill in the blanks for each of the following angle measures: m m m. What type of triangle is Δ? (lassify by both angles and sides) How do you know? 4. Now, focusing on just Δ taken from square, use the Pythagorean Theorem to find each of the missing side lengths in the following right isosceles triangles. Make sure to epress sides as simplified radicals (for instance, a b ), not as decimals. 1 = 1, =, = 2 = 2, =, = =, =, = =, =, = 8 = 8, =, = =, =, = 5. These triangles are all known as right isosceles triangles, but are also referred to as triangles because of the measures of their angles. What pattern do you notice about the legs and hypotenuse of any triangle?. How could you use this pattern to help you find the missing sides of any triangle without having to use the Pythagorean Theorem? Special Right Triangles p. 1

6 1. Label the missing angle measures for all angles in this equilateral triangle, and mark any congruent sides with congruency marks. 2. ill in the blanks for each of the following angle measures: m EH m HE m HE. What type of triangle is ΔEH? (lassify by both angles and sides) How do you know? 4. What kind of special segment is H in ΔEG? What does H do to EG? What is the relationship between m E and m EH? 5. Now, focusing on just ΔEH taken from equilateral ΔEG, use the information you know about how special segment H interacts with EG and the Pythagorean Theorem to find each of the missing side lengths in the following triangles. Make sure to epress the sides as simplified radicals (for instance, a b ), not as decimals. E H G 4 E H E EH =, H =, E = 4 EH =, H =, E = H 10 E H E 7 EH =, H =, E = 10 H EH = 7, H =, E = E 8 H EH = 8, H =, E = EH =, H =, E =. These triangles are usually referred to as triangles, because of the measures of their angles. The short leg is across from the 0, the long leg is across from the 0 angle, and the hypotenuse is across from the 90 angle. What pattern do you notice eists between the short leg, long leg, and hypotenuse of any triangle? E H 7. How could you use this pattern to help you find the missing sides of any triangle without having to use the Pythagorean Theorem? Special Right Triangles p. 2

7 Geometry Notes Intro to Trig, and Solving Trig Problems Name ate is the study involving right triangles. is a ratio of the lengths of 2 sides in a right triangle. When we apply this to a RIGHT TRINGLE, you must look for the (sometimes called theta θ ). We have common Trig functions: Sine ( ) is LWYS osine ( ) is LWYS Tangent ( ) is LWYS θ c b a Now we are going to look at a way to remember the ratios. Sin Opposite Hypotenuse os djacent Hypotenuse Tan Opposite djacent So how do we use this? SOHHTO tells you which sides to use in relation to the angle you are looking at. ***Step by Step Method for Solving Trig Problems*** 1. Write your variables (θ, opp, adj, and hyp) 2. Pick your function (not your nose!). Write your ratio using your variables. 4. Solve for. * heck that your calculator MOE is in EGREE!! EX 1 ind 5 Step 1 Step 2 Step 4 θ = opp = adj = Step hyp = EX 2 ind 1 47 Step 1 Step 2 Step 4 θ = opp = adj = Step hyp = EX ind Step 1 Step 2 Step 4 24 θ = opp = 2 adj = Step hyp =

8 Geometry Notes Solving Inverse Trig Problems Name ate We have used trig ratios to find the measure of a missing side. Now we are going to use it to find the measure of a angle. We will use the same 4 steps to set up the problems, but we will be using our inverse trig functions to help use solve them. We do this by pressing 2 nd then sin, cos, or tan on our calculator. Sin Opposite Hypotenuse os djacent Hypotenuse Tan Opposite djacent ***Step by Step Method for Solving Trig Problems*** 1. Write your variables (θ, opp, adj, and hyp) 2. Pick your function (not your nose!). Write your ratio using your variables. 4. Solve for. Lets do a few problems together: EX 1 ind 5 14 Step 1 Step 2 Step 4 θ = opp = adj = Step hyp = EX 2 ind 12 1 Step 1 Step 2 Step 4 θ = opp = adj = Step hyp = EX ind 2 24 Step 1 Step 2 Step 4 θ = opp = adj = Step hyp =

9 Geometry Pre-P WS Trig with a alculator Name ate Solve for in each of these problems. Remember to look at your eample sheet to help you. lso, remember SOHHTO. Round to decimals places (nearest thousandth), for both side lengths and angle measures. The P tests uses significant figures. heck that the calculator MOE is in EGREE = = = = = =

10 = = = = = =

11 Name ate Period Notes - ngles of Elevation and epression Many problems in daily life can be solved by using trigonometry. Often such problems involve an angle of elevation or an angle of depression. Eample: The angle of elevation from point to the top of a cliff is 8. If point is 80 feet from the base of the cliff, how high is the cliff? Let represent the height of the cliff. Solve each problem. Round measures of segments to the nearest hundredth and measures of angles to the nearest whole degree. 1. rom the top of a tower, the angle of depression to a stake on the ground is 72. The top of the tower is 80 feet above ground. How far is the stake from the foot of the tower? 2. tree 40 feet high casts a shadow 58 feet long. ind the measure of the angle of elevation of the sun.. ladder leaning against a house makes an angle of 0 with the ground. The foot of the ladder is 7 feet from the foundation of the house. How long is the ladder? 4. balloon on a 40-foot string makes an angle of 50 with the ground. How high above the ground is the balloon if the hand of the person holding the balloon is feet above the ground?

12 Right Triangle ctivity Name ate Pd nswer enough problems correctly to earn the grade you would like on this activity. e sure to show all of your work and circle/bo your answer pts 2. 5 pts. 5 pts 4. 5 pts 5. 5 pts. 5 pts 7. 5 pts 8. 5 pts

13 9. 10 pts pts pts pts pts pts