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Featured 4 months ago

Some of the properties of a parallelogram:

a) Opposite sides are parallel.

b) Opposite sides are congruent.

c) Opposite angles are congruent.

d) The diagonals bisect each other,

The Midpoint Formula: The midpoint of two points,

First, let

Hence, the first coordinates of vertex

Then, let

So the second coordinates for vertex Y are

Hence, the two possible coordinates for vertex Y are

Featured 2 months ago

Area of the region outside the house that the dog can reach is

As the dog is tied to a corner of a hexagon, with a rope of length of

As may be seen, he has freedom of movement in a circle of

Hence the area of the region outside the house that the dog can reach is given by

=

=

=

=

=

Featured 2 months ago

The equation of the circle is

The standard Cartesian form for the equation of a circle is:

where

Given: The tangent line

Because the radius is the circle is perpendicular to the tangent, we can use the tangent line to find an equation that pass through the center. Write the tangent line in slope-intercept form:

This makes the center line:

Evaluate this at the point

Evaluate equation [1] and the points

Expand the squares:

Subtract equation [5] from equation [6]:

Substitute into equation [2]:

Substitute into equation [5]:

The equation of the circle is

Here is a graph of, the circle (red), the center point (blue), the tangent line (green) and the center line (yellow):

Featured 2 months ago

Area

When a line from one of the vertex of a triangle, meets the opposite side, the ratio between the areas of the two triangles is same as the ratio of the two parts in which it divides the opposite side.

By corollary, in a quadrilateral whose diagonals divide it in four triangles, areas of two adjacent triangles are in the same ratio as the ratio of other two triangles.

Let

Given

As

let

As

Area of triangle

Featured 1 month ago

see explanation.

1.

Area of a cricle

circumference of a circle

2.

Let

As shown in the figure, the inscribed circle has a diameter

Let

3.

The largest circle that can fit insides a square of edge of

Area of circle

Let

Area of the materail wasted = shaded area =

4.

Area of washer (one face) = shaded area

given

5.

An equilateral triangle is a triangle in which all 3 sides are equal and all 3 angles are each

As two tangents to a circle from an external point are equal,

As

annulus area = shaded area

Featured 2 weeks ago

Let the equation of the circle be

where

Let

So the equation of the tangent on the circle at

So **the equations of set of chords parallel to that tangent** can be written as

where c is another parameter which changes from chord to chord.

Multiplying {1} by

From [2] and [3] we can write

If

If

Now

So

Hence

Now equation of the diameter of the circle passing through

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