Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Line (geometry) wikipedia , lookup
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Multilateration wikipedia , lookup
Euler angles wikipedia , lookup
Trigonometric functions wikipedia , lookup
Geometry Problems: Angles, Triangles, Perimeter Basic facts you need to know Complementary angles add to 90°, same as a right angle. Supplementary angles add to 180°, same as a straight line angle. Page 1 of 2 Complementary Angles add to 90° The problem. Find two complementary angles such that the measure of the first angle is 15° less than twice the measure of the second angle. Supplementary Angles add to 180° The problem. Find two supplementary angles such that the measure of the larger angle is 110° more than triple the measure of the smaller angle. 1. Thinking about it. We have a clue about the first angle but no clue about the second angle. 1. Thinking about it. We have a clue about the larger angle but no clue about the smaller angle. 2. The dictionary (write it out!). Let π₯ = the second angleβs measure. Then 2π₯ β 15 = the first angleβs measure. 2. The dictionary (write it out!). Let π₯ = the smaller angleβs measure. Then 3π₯ + 110 = the larger angleβs measure. 3. The equation. π₯ + 2π₯ β 15 = 90 The 90 is because of complementary angles. 3. The equation. π₯ + 3π₯ + 110 = 180 The 180 is because of supplementary angles. 4. Solve the equation. Combine like terms: 3π₯ β 15 = 90. +15 to each side: 3π₯ = 105 Divide each side by 3: π₯ = 35. 4. Solve the equation. Combine like terms: 4π₯ + 110 = 180. -110 to each side: 4π₯ = 70 Divide each side by 4: π₯ = 17.5 (In this special case the decimal will be okay.) 5. Bring the solution back to the dictionary. Let π₯ = the smaller angleβs measure = 17.5°. Then 3π₯ + 110 = the larger angleβs measure = 3(17.5) + 110 β 162.5° 5. Bring the solution back to the dictionary. Let π₯ = the second angleβs measure. = 35° Then 2π₯ β 15 = the first angleβs measure = 2(35) β 15 β 70 β 15 β 55°. 6. Answer the question they asked. The first angle measures 55° and the second angle measures 35°. 6. Answer the question they asked. The angles measure 17.5° and 162.5°. Quick check. Add: 17.5+162.5=180; that Quick check. Add: 55+35=90; that agrees agrees with the definition of supplementary with the definition of complementary angles. angles. Check your solution against the original problem: does it work? And always be sure to answer the question they asked! D.R.S., 10/18/2013 12:26 PM Geometry Problems: Angles, Triangles, Perimeter Three angles of a triangle sum to 180° The problem. The smallest angle of a triangle measures one-fifth as much as the largest angle. The other angle measures ten degrees more than one-half of the largest angle. Find the anglesβ measurements. 1. Thinking about it. We have clues about the smallest angle and the second angle. There arenβt any clues given about the largest angle Page 2 of 2 Perimeter of a rectangle The problem. A rectangular lot of land has a length that is 50 feet shorter than three times its width. Its perimeter is 460 feet. What are the dimensions of the lot? 1. Thinking about it. We have a clue about the length. 2. The dictionary (write it out!). Let π₯ = the largest angle. 1 Then π₯ = the smallest angle. 2. The dictionary is a picture. Draw it. Label the unknown width as x. Label the length according to the clue β50 feet shorter than the width.β And + 10 = the second angle. 3. The equation. The key fact to know is that the three angles of a 1 1 triangle sum to 180° π₯ + π₯ + π₯ + 10 = 180 5 2 4. Solve the equation. Clear the fractions first. LCD of 5 and 2 is 10, so multiply each term by 10: 10π₯ + 2π₯ + 5π₯ + 100 = 1800 Combine like terms: 17π₯ + 100 = 1800 Then -100 on each side: 17π₯ = 1700 Div each side by 17: π₯ = 100 3. The equation. The key fact to know is that perimeter is the distance around the sides. Add up all the sides = the perimeter. 3π₯ β 50 + π₯ + 3π₯ β 50 + π₯ = 460 4. Solve the equation. Combine like terms: 8π₯ β 100 = 460 Then +100 on each side: 8π₯ = 560 Div each side by 8: π₯ = 70 5 1 π₯ 2 5. Bring the solution back to your diagram; plug in for x. 5. Bring the solution back to the dictionary. Let π₯ = the largest angle = 100° 1 1 Then 5 π₯ = the smallest angle = 5 (100) β20° 1 2 And π₯ + 10 = the second angle= 1 (100) + 2 10 β60° Quick check. Yes, 160 + 70 + 160 + 70 = 460 feet. Quick check. Yes, 100 + 20 + 60 = 180 6. Answer the question they asked. (Read carefully to put the correct angle value in the correct box in the MyMathLab questions! ) 6. Answer the question they asked. (Read carefully to put the correct angle value in the correct box in the MyMathLab questions!) D.R.S., 10/18/2013 12:26 PM