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## Saturday, 9 August 2014

### chapter 4 Areas and Perimeters

Theorem $14$
The area of a parallelogram is equal to $bh$ where $b$ is the length of its base and $h$ is its height (the perpendicular distance from the base to the parallel side opposite).
Theorem $15$
The area of a triangle is equal to $\dfrac{1}{2}bh$ where $b$ is the length of its base and $h$ is its height (the perpendicular distance from the base to the vertex opposite).
Theorem $16$ Heron’s Theorem
If the lengths of the sides of a triangle are $a$$b$ and $c$, so that the semiperimeter $s = (a + b + c)/2$ then the area of the triangle is
 $\sqrt {s(s - a)\;(s - b)\;(s - c)}$
Circles
Theorem $17$
The area of a circle of radius $r$ is $\pi {r^2}$ and its circumference is $2\pi r$.
Theorem $18$
If $AB$ is an arc of a circle then angles subtended at the circumference opposite $AB$ are equal and are equal to half the angle subtended at the centre, i.e. in the diagram $\angle ACB = \angle ADB = \dfrac{1}{2}\angle \,AOB$.
Theorem $19$
If $AB$ is diameter of a circle and $C$ is any point on the circumference of the circle then $\angle ACB$ is a right angle.
Theorem $20$
If $A$ and $B$ are points on the circumference of a circle with centre $O$ and $C$ is an exterior point of the circle such that $BC$ is a tangent to the circle then $\angle ABC = \dfrac{1}{2}\angle AOB$.
Theorem $21$
A line from the centre of a circle perpendicular to a chord bisects the chord and its arc.
Theorem $22$
If $A$ is a point on the circumference of a circle then a tangent to the circle at $A$is perpendicular to a radius of the circle to $A$.
Theorem $23$
The two tangents drawn to a circle drawn from an exterior point of the circle have the same length.
Theorem $24$
If two circles touch at a single point then this point and the centres of the circles are collinear.
Theorem $25$
If two circles intersect at two points then the line through their centres is the perpendicular bisector of their common chord.
Theorem $26$
Opposite angles of a cyclic quadrilateral sum to ${180^ \circ }$ and if a pair of opposite angles of a quadrilateral sum to ${180^ \circ }$ then it is cyclic.
Theorem $27$
If $XE$ is a tangent to the circle ($X$ being any point on the plane of the circle) and $A$$B$$C$$D$ are points on the circumference, we have the relations $XA \cdot XB = XC \cdot XD = X{E^2}$.

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