ONE MORE STEP TOWARD BETTER TOMORROW : chapter 4 Areas and Perimeters

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 $XA \cdot XB = XC \cdot XD = X{E^2}$ .

Saturday, 9 August 2014

chapter 4 Areas and Perimeters

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