Study Mathematics notes on Percentages, Congruence and Similarity, Algebraic Identities, Quadratic Formula for O-Level. Useful for revision, homework, and exam preparation in Zimbabwe.
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This is the O-Level Mathematics notes. It is a free notes for revision.
Important points and formulae
1. Percentages:
Increase = new value - original value
Decrease = original value - new value
Profit $=$ selling price - cost price
Loss $=$ cost price - selling price
Discount $=$ marked price - sale price
\[
\begin{array}{l}
\% \text { discount }=\frac{\text { discount }}{\text { markedprice }} \times 100 \%
\% \text { profit } / \text { loss }=\frac{\text { profit } / \text { loss }}{\text { originalprice }} \times 100 \%
\end{array}
\]
2. Congruence and similarity
Two triangles are congruent if
a) All sides of one triangle are equal to all sides of another triangle. (SSS property)
b) Two sides and including angle of one triangle are equal to two corresponding sides and including angle of another triangle. (SAS property)
c) Two angles and one side of one triangle are equal to two corresponding angles and one side of another triangle. (AAS property)
d) A figure and its image under translation, rotation and reflection are congruent.
When two polygons are similar, then
a) All corresponding angles are equal,
b) All corresponding sides are proportional.
c) A figure and its image under an enlargement are similar.
d) An enlargement with a scale factor greater than 1 produces an enlarged figure. An enlargement with a scale factor between 0 and 1 produces a diminished image. An enlargement with a scale factor 1 produces a congruent image.
e) Two triangles are said to be similar if, All three angles of one triangle are equal to all corresponding angles of another triangle. (AA property)
One angle of both the triangles is equal and two corresponding sides are proportional. (ASS property)
f) If two shapes are similar,
\[
\frac{A 1}{A 2}=\left(\frac{l 1}{l 2}\right) 2
\]
g) If two shapes are similar,
\[
\frac{v 1}{v 2}=\left(\frac{l 1}{l 2}\right) 3
\]
h) If XY is parallel to BC then $\mathrm{AX} / \mathrm{XB}=\mathrm{AY} / \mathrm{YC}$ (see fig)
4. Algebraic identities
a) $(a+b)^{2}=a^{2}+2 a b+b^{2}$,
b) $(a-b)^{2}=a^{2}-2 a b+b^{2}$,
c) $(a+b)(a-b)=a^{2}-b^{2}$
Quadratic formula