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The height is often a line drawn inside the triangle. This is the case in four of the six triangles above. But if the triangle is right-angled, the height can be one of the sides . This can be seen in the fourth triangle. In the sixth triangle you can see that the height line needs to be drawn outside the triangle.
Summary:
In summary, if you want to use the area formula you need to have a base and a height that make a pair, and you must have (or be able to calculate) their lengths. In some of the following problems, you will have to calculate the area of a triangle on the way to an answer.
Here is a reminder of the Theorem of Pythagoras; it applies only to right-angled triangles, but you will encounter many of those from now on.
I
In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
If you are a bit vague about applying the theorem, go back to the work you did on it before and refresh your memory.
A = 2 × area of 1 triangle = 2 (½ × base × height) = 2 × ½ × s × s = s ^{2} = side squared.
You probably knew this already!
A = first triangle + second triangle
= (½ × base × height) + (½ × base × height)
= (½ × b × $\ell $ ) + (½ × $\ell $ × b) = ½ b $\ell $ + ½ b $\ell $ = b $\ell $
= breadth times length.
You probably knew this already!
A = triangle _{1} + triangle _{2} = ½ × P s _{1} × h + ½ × P s _{2} × h
= ½ h ( P s _{1} + P s _{2} ) = half height times sum of parallel sides.
(Did you notice the factorising?)
Area = 2 identical triangles
= 2( ½ × sl × h ) = 2 × ½ × sl × ½ × sd
= sl × ½ × sd = ½ × sl × sd
= half long diagonal times short diagonal.
In the following exercise the questions start easy but become harder – you have to remember Pythagoras’ theorem when you work with right angles.
Calculate the areas of the following quadrilaterals:
1 A square with side length 13 cm
2 A square with a diagonal of 13 cm (first use Pythagoras)
3 A rectangle with length 5 cm and width 6,5 cm
4 A rectangle with length 12 cm and diagonal 13 cm (Pythagoras)
5 A parallelogram with height 4 cm and base length 9 cm
6 A parallelogram with height 2,3 cm and base length 7,2 cm
7 A rhombus with sides 5 cm and height 3,5 cm
8 A rhombus with diagonals 11 cm and 12 cm
(What fact do you know about the diagonals of a rhombus?)
9 A trapezium with the two parallel sides 18 cm and 23 cm that are 7,5 cm apart
10 A kite with diagonals 25 cm and 17 cm
LO 3 |
Space and Shape (Geometry)The learner will be able to describe and represent characteristics and relationships between two-dimensional shapes and three–dimensional objects in a variety of orientations and positions. |
We know this when the learner: |
3.2 in contexts that include those that may be used to build awareness of social, cultural and environmental issues, describes the interrelationships of the properties of geometric figures and solids with justification, including: |
3.2.2 transformations. |
3.3 uses geometry of straight lines and triangles to solve problems and to justify relationships in geometric figures; |
3.4 draws and/or constructs geometric figures and makes models of solids in order to investigate and compare their properties and model situations in the environment. |
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