For the figure below, we are given that
$BC=BD=x$ .
Show that
$B{C}^{2}=2{x}^{2}(1+sin\theta )$ .
We want
$CB$ , and we have
$CD$ and
$BD$ . If we could get the angle
$B\widehat{D}C$ , then we could use the cosine rule to determine
$BC$ . This is possible, as
$\u25b5ABD$ is a right-angled triangle. We know this from circle geometry, that any triangle circumscribed by a circle with one side going through the origin, is right-angled. As we have two angles of
$\u25b5ABD$ , we know
$A\widehat{D}B$ and hence
$B\widehat{D}C$ . Using the cosine rule, we can get
$B{C}^{2}$ .
Using the above, show that
$sin2\theta =2sin\theta cos\theta $ .
Now do the same for
$cos2\theta $ and
$tan\theta $ .
$DC$ is a diameter of circle
$O$ with radius
$r$ .
$CA=r$ ,
$AB=DE$ and
$D\widehat{O}E=\theta $ .
Show that
$cos\theta =\frac{1}{4}$ .
The figure below shows a cyclic quadrilateral with
$\frac{BC}{CD}=\frac{AD}{AB}$ .
Show that the area of the cyclic quadrilateral is
$DC\xb7DA\xb7sin\widehat{D}$ .
Find expressions for
$cos\widehat{D}$ and
$cos\widehat{B}$ in terms of the quadrilateral sides.
Show that
$2C{A}^{2}=C{D}^{2}+D{A}^{2}+A{B}^{2}+B{C}^{2}$ .
Suppose that
$BC=10$ ,
$CD=15$ ,
$AD=4$ and
$AB=6$ . Find
$C{A}^{2}$ .
Find the angle
$\widehat{D}$ using your expression for
$cos\widehat{D}$ . Hence find the area of
$ABCD$ .
Problems in 3 dimensions
$D$ is the top of a tower of height
$h$ . Its base is at
$C$ . The triangle
$ABC$ lies on the ground (a horizontal plane). If we have that
$BC=b$ ,
$D\widehat{B}A=\alpha $ ,
$D\widehat{B}C=x$ and
$D\widehat{C}B=\theta $ , show that
$$h=\frac{bsin\alpha sinx}{sin(x+\theta )}$$
We have that the triangle
$ABD$ is right-angled. Thus we can relate the height
$h$ with the angle
$\alpha $ and either the length
$BA$ or
$BD$ (using sines or cosines). But we have two angles and a length for
$\u25b5BCD$ , and thus can work out all the remaining lengths and angles of this triangle. We can thus work out
$BD$ .
The line
$BC$ represents a tall tower, with
$C$ at its foot. Its angle of elevation from
$D$ is
$\theta $ . We are also given that
$BA=AD=x$ .
Find the height of the tower
$BC$ in terms of
$x$ ,
$tan\theta $ and
$cos2\alpha $ .
Find
$BC$ if we are given that
$x=140m$ ,
$\alpha ={21}^{\circ}$ and
$\theta ={9}^{\circ}$ .
Other geometries
Taxicab geometry
Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates.
Manhattan distance
The metric in taxi-cab geometry, is known as the
Manhattan distance , between two points in an Euclidean space with fixed Cartesian coordinate system as the sum of the lengths of the projections of the line segment between the points onto the coordinate axes.
For example, the Manhattan distance between the point
${P}_{1}$ with coordinates
$({x}_{1},{y}_{1})$ and the point
${P}_{2}$ at
$({x}_{2},{y}_{2})$ is
Questions & Answers
I only see partial conversation and what's the question here!
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?