Two line segments are divided in the
same proportion if the ratios between their parts are equal.
If the line segments are proportional, the following also hold
and
and
Proportionality of triangles
Triangles with equal heights have areas which are in the same proportion to each other as the bases of the triangles.
A special case of this happens when the bases of the triangles are equal:
Triangles with equal bases between the same parallel lines have the same area.
Triangles on the same side of the same base, with equal areas, lie between parallel lines.
Theorem 1 Proportion Theorem: A line drawn parallel to one side of a triangle divides the other two sides proportionally.
Given :
ABC with line DE
BC
R.T.P. :
Proof :
Draw
from E perpendicular to AD, and
from D perpendicular to AE.
Draw BE and CD.
Similarly,
Following from Theorem
"Proportion" , we can prove the midpoint theorem.
Theorem 2 Midpoint Theorem: A line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side.
Proof :
This is a special case of the Proportionality Theorem (Theorem
"Proportion" ).
If AB = BD and AC = AE,and
AD = AB + BD = 2ABAE = AC + CB = 2AC
then DE
BC and BC = 2DE.
Theorem 3 Similarity Theorem 1: Equiangular triangles have their sides in proportion and are therefore similar.
Given :
ABC and
DEF with
;
;
R.T.P. :
Construct: G on AB, so that AG = DE, H on AC, so that AH = DF
Proof :
In
's AGH and DEF
means “is similar to"
Theorem 4 Similarity Theorem 2: Triangles with sides in proportion are equiangular and therefore similar.
Given :
ABC with line DE such that
R.T.P. :
;
ADE
ABC
Proof :
Draw
from E perpendicular to AD, and
from D perpendicular to AE.
Draw BE and CD.
Theorem 5 Pythagoras' Theorem: The square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides.
Given :
ABC with
Required to prove :
Proof :
In
GHI, GH
LJ; GJ
LK and
=
. Determine
.
We need to calculate
:
We were given
So rearranging, we have
And:
Using this relation:
PQRS is a trapezium, with PQ
RS. Prove that PT
TR = ST
TQ.
Triangle geometry
Calculate SV
. Find
.
Given the following figure with the following lengths, find AE, EC and BE.
BC = 15 cm, AB = 4 cm, CD = 18 cm, and ED = 9 cm.
Using the following figure and lengths, find IJ and KJ.
HI = 26 m, KL = 13 m, JL = 9 m and HJ = 32 m.
Find FH in the following figure.
BF = 25 m, AB = 13 m, AD = 9 m, DF = 18m.
Calculate the lengths of BC, CF, CD, CE and EF, and find the ratio
.
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,