Download Revision Aid: Zeta Club Factsheet - Geometry

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Transcript
Junior/Intermediate Geometry Tips
Angles
1. Make sure you can quickly apply the basics, i.e. interior angles of triangle add up to 180°,
interior angles of quadrilateral add up to 360°. The one below is often easy to miss:
2. Exterior angle of a regular polygon with n sides (between an edge and another edge
β€˜extended out’):
360
𝑛
Therefore interior angle = 180 βˆ’
360
𝑛
3. Spot when you have an isosceles triangle, when you know two of the lengths of the triangle
must be the same. Then two of the angles must also be the same.
4. Circle Theorems: (which you’re unlikely to need until the Senior Maths Challenge/Olympiad)
A tangent is a straight line that touches a circle.
The inscribed
angle is half
the central
angle.
Alternate Segment
Theorem: The angle
between a chord
and tangent is equal
to the angle in the
alternate segment.
The angle inscribed in a
semicircle is always 90°.
If the radius of a circle
and a tangent touch,
the angle between
them is 90°.
5. Look for opportunities to draw a circle in the diagram (e.g. around a regular polygon). This
allows you to take advantage of circle theorems.
Areas and Lengths
1. In many problems you’ll have a right-angled isosceles triangle, i.e. with angles 90°, 45°, 45°.
a. If you know the length of the hypotenuse: Divide by √2 to get the length of the
smaller sides.
b. If you know the length of the shorter sides: Multiply by √2 to get the length of the
hypotenuse
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2. Look for opportunities to cut up the area of a shape, to either divide it into shapes you can
calculate the area for, or rearrange the bits to form a new simpler shape.
π‘Ž
𝑏
3. For two similar triangles: =
𝑐
𝑑
4. Introduce variables for unknown lengths (or angles).
5. In general, look out for opportunities to form equations when you have more complicated
expressions involving unknown quantities. For example, this might be because you formed a
right-angled triangle and want to use Pythagoras Theorem.
6. Particularly for problems involving multiple circles or circles inscribed in other shapes, add
extra lines to your diagram, e.g. the radius of your circle, in such a way that it helps you form
an expression or equation.
7. And in general add extra lines to your diagram! These might be lengths that you already
know, ones which you can give some expression for in terms of existing variables (e.g. √2π‘Ÿ)
or ones that you don’t yet know – in which case introduce a variable.
πœƒ
8. Area of sector: 360 × πœ‹π‘Ÿ 2
9. For any problem involving the area of a shape with an arc for one edge and the other edges
straight (e.g. a circle segment), start with a sector of the corresponding circle, then β€˜cut out’
any polygonal areas as necessary. For example, in the problem below where we need to
establish the area of the shaded region, we might divide it up into four so each region has
only one curved edge, then subtract areas as follows:
10. When areas are stated to be equal, be methodical in working out the two areas, equating
them, and simplifying.
11. Be careful about using radius vs diameter, and remembering to half your area for semicircles, etc.
12. It’s worthwhile memorising this table:
sin
45°
1
cos
√2
1
tan
√2
1
30°
1
2
√3
2
1
√3
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60°
√3
2
1
2
√3