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Solving triangles

Solving triangles. Your two triangles are the same shape and size. They are congruent. These two triangle have the same shape but. are different sizes. They are similar triangles. They have the same angles. Problems involving triangles can be solved by scale drawing. See examples.

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Solving triangles

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  1. Solving triangles

  2. Your two triangles are the same shape and size They are congruent

  3. These two triangle have the same shape but are different sizes They are similar triangles They have the same angles

  4. Problems involving triangles can be solved by scale drawing See examples Problems involving triangles can be solved by calculation using geometry See examples Problems involving triangles can be solved by calculation using trigonometry See examples

  5. Scale drawings are similar to the real thing

  6. North The map shows the journey of a ship. Island The ship leaves the Port, sails 30 miles North, then 20 miles East. MAP N Make a scale drawing & produce a similar diagram. I How far is the Island from the port ? ? Port DRAWING P

  7. Geometry uses rules that are true for all triangles

  8. What is the size of the green angle in this triangle ? Answer 50º 55º 75º Use Geometry

  9. Pythagoras of Samos • Born: about 569 BC • in Samos, Ionia • Died: about 475 BC

  10. Pythagoras’ Theorem is about triangles Right angled triangles Draw a square onto each side The area of the largest square equals the area of the two smaller squares See Examples

  11. A Calculate the length of side AB x cm 6 cm B C 8 cm By Pythagoras’ theorem x2 =6 2 + 82 = 36 + 64 = 100 x = 10 The length of side AB is 10 cm

  12. T Calculate the length of side RT y m R 15 m 17 m By Pythagoras’ theorem 17 2= y 2+ 15 2 289= y 2+ 225 y 2 = 64 S y = 8 The length of side RT is 8 m

  13. Trigonometry uses facts about similar triangles

  14. Try this experiment Cut the other piece in half like this. Take a sheet of A4 paper this way up. Cut it in half   Repeat as often as you can so you get a sequence of rectangles

  15. Place all the rectangles on top of each other like this Draw a diagonal line Cut along the diagonal line to make lots of right angled triangles  Which triangles are congruent? Which are similar?

  16. All these right angled triangles are similar. These triangles are the same shape but different sizes. Any triangle similar to these has the same angles

  17. On card make some right angled triangles that are: twice as long as they are high See if all your triangles are similar. Measure the smallest angle

  18. We can use trigonometry to compare similar triangles We can use scientific calculator to give us information about right angles triangles of all different shapes & sizes.

  19. H To calculate the angle x O x You need to label the sides H = hypotenuse (longest side, opposite the right angle) O = opposite (opposite the angle x) A = adjacent (next to the angle x) A Then you have to choose between these ratios OH sine x = (Oranges Have Segments) AH cosine x = (Apples Have Cores) tangent x = OA (Oranges Are Tasty)

  20. To calculate the angle x H O 10 cm x Label the sides (H O A) 20 cm A Choose: sin, cos or tan? The two sides involved are O & A OA tan x = (Oranges Are Tasty) 1020 tan x = tan x = 0.5 Using the calculator tan-1 function x = 26.6º to 1 decimal place

  21. H To calculate the value of y 10 cm y cm Label the sides (H O A) 30º O Choose: sin, cos or tan? The two sides involved are O( y cm) & H(10 cm) A OH sin 30º = (Oranges Have Segments) sin 30º = Y 10 y = 10 x sin 30º Using the calculator sinefunction x = 5 cm

  22. 12 mm A To calculate the value of h 52º O Label the sides (H O A) Choose: sin, cos or tan? The two sides involved are A & H h mm H AH cos 52º = (Apples Have Cores) 12 h cos 52º = 12___ cos 52º h = Using the calculator cosinefunction h = 19.5 mm to 3 significant figures

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