Triangles

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Working with triangles

 When using triangles it is possible to work out the length of a side of a triangle or the size of an angle within it if you know some of the other sides or angles. The study of the relationship between the length of sides and the size of angles is called trigonometry.

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Types of triangle are classified in terms of their sides and angles, to find out about the individual types of triangle click on the links below:

There is help on working out the value of an angle in any triangle when you know the other two.

If working with right-angled triangles, there is help with:

Types of triangle

 Equilateral triangle. A triangle with all three sides the same length is called an equilateral triangle. If all the sides are the same length, then all the angles must also be the same. In an equilateral triangle each angle equals 60° (180 ¸ 3). Isosceles triangle. An isosceles triangle has two sides the same length and two angles the same. The two angles at the 'feet' of these sides are equal, ie the angles that are not between the sides of the same length (see left). Scalene triangle. A scalene triangle has no sides or angles that are the same. Right-angled triangle. A right-angled triangle has one angle equal to 90°. the right-angle in the triangle is marked with a 'square' symbol. (see left). A right-angled triangle can also be either isosceles or scalene.

Value of an angle in any triangle when you know the other two

 The angles within a triangle must always add up to 180 degrees. A + B + C= 180 So if you know the value of any two angles it is possible to work out the value of the third eg: A = 180 - B - C Example 1 So if A = 180 - B - C and B is 100° and C is 30° then A = 180 - 100 - 30 So A = 50° Right-angled triangle. The right angle in a triangle is marked with a square rather than an arc (see C left). This tells you that the angle is a right angle, ie it is 90 degrees. So in this case you only need to know one of the other angles, as C is given by its notation. A = 180 - B - 90 this is the same as A = 90 - B and also B = 90 - A This is because we know that 90 of the 180 degrees in the triangle are angle C. So the remaining 90 degrees must be made up of the other two. Example 2 So A = 90 - B, where B is 36° then A = 90 - 36 So A = 54 ° When a triangle has sides that are same length. A triangle with all three sides the same length is called an equilateral triangle. If all the sides are the same length, then all the angles must also be the same. In an equilateral triangle each angle is 60 degrees (180 ¸ 3) When a triangle has two sides the same length it is called an isosceles triangle. The two angles at the 'feet' of these sides are also equal, ie the angles that are not between the sides of same length (see left) The coloured marks on the diagrams left show that the sides and angles are the same. Example 3 So if A = 180 - B - C and C is 72° then A = 180 - B - 72 so A = 108 - B this is the same as B = 108 - A and as A + B = 108 If A + B = 108 and A and B are the same (as it is an isoscoles triangle) then A = 108/2 so A = 54 ° In summary we can state that the angles in the isosceles triangle are A = 54 degrees, B = 54 ° (being the same as A) and C = 72 ° Word problems. Sometimes you may need to work out the value of an angle within a word problem, without a picture of the triangle in question. eg In a traingle ABC what is the value of A when C is 72 °? Angles A and B are equal. This is the same question as example 3 above. It might help you to work out the angle by doing a rough sketch of the triangle and giving the angles letters and marking the angles that are the same. Activity 1 If you want to practice finding the missing angle in a triangle, have a go at the questions below In a traingle ABC, what is the value of A where B = 56° and C = 84°? In a triangle DEF, what is the value of E where D = 25° and F = 37°? In a right-angled triangle GHI, what is the value of G where H is the right angle and I is 48°? In an isosceles triangle JKL, what is the value of J where K = 71°? K an L are equal. In a triangle MNP, what is the value of P where M = 112° and N and P are equal? Are you right? If not, have another look at the example earlier in this topic or have a look at some of the 'Resources You Can Use' that are recommended for this topic.

Finding sides and angles of a right-angled triangle

 Finding the length of one side when you know the other two. There is a relationship between the length of the sides of any right-angled triangle. This is often referred to as Pythagoras' theorem. The theorem (or relationship) is named after Pythagoras, a Greek who lived around 500BC. Although the relationship was known long before Pythagoras lived, he is thought to be responsible for proving that the relationship ALWAYS exists for any size right-angled triangle. Pythagoras' theorem. The relationship can be described as: "the square on the hypotenuse is equal to the sum of the squares on the other two sides" This means that if the length of each side of the triangle is made into a square, the area of the square on the longest side is equal to the areas of the squares on the other two sides added together. The hypotenuse. The hypotenuse is the name for the longest side. The hypotenuse is also always the one directly opposite the right angle. Pythagoras' theorem can also be written as a formula. a2 + b2 = c2 see left Example 4 In a right-angled triangle, what is the length of side c if a = 3 and b = 4? Given a2 + b2 = c2 so 32 + 42 = c2 9 + 16 = c2 25 = c2 c=Ö25 So c = 5 For more help with squares and square roots see the sub topic 'Powers and Roots' in the menu on the left of the screen. Example 5 In a right-angled triangle, what is the length of side b if a = 5 and c = 13? Given a2 + b2 = c2 so 52 + b2 = 132 25 + b2 = 169 Now rearrange the formula so that b2 is on its own on the left hand side. You can do this by taking away 25 from the left hand side. Anything that you do to one side, you must do to the other side so the equation still balances. So take 25 away from the right hand side too. For more help with rearranging equations see the sub topic 'Formulae and algebra' in the menu on the left of the screen. 25 + b2 - 25 = 169 - 25 b2 = 169 - 25 b2 = 144 b = Ö144 So b = 12 Activity 2 If you want to practice finding the length of a side of a right-angled triangle using Pythagoras' theorem, have a go at the questions below: For questions 1-4 use the triangle ABC left as a guide. Produce all answers to 2 decimal places: What is the length of c where a = 7 and b = 24? What is the length of c where a = 5 and b = 11? What is the length of a where b = 8 and c = 30? What is the length of b where a = 2 and c = 7? An isosceles triangle has a hypotonuse of 18 cm, what are the lengths of the other two sides? Are you right? If not, have another look at the example earlier in this topic or have a look at some of the 'Resources You Can Use' that are recommended for this topic.

Finding the length of one side when you know the length of one other side and the size of one angle