We will be covering geometry and arithmetic.
Two properties that will help you in determining the length given the angle or
determining the angle given lengths are the sine rule and cosine rule. While
they are not absolute must-knows, they can significantly reduce the
complexity of questions that are difficult.
As the angle grows, the length of the side of a triangle corresponding to that
angle increases proportionally. The proportionality theorem states that
suppose there is a triangle ABC, and you draw a line DE parallel to BC, then DE
will split AB in the same ratio as it splits AC. This ratio is the ratio of the length
of the part of AD to DB, and AE to EC.
One way to solve questions on triangles is by using the basic theorem where
one side of a smaller triangle is parallel to one side of a bigger triangle. By
using this, the ratios of the areas and lengths of the sides can be figured out.
Another useful tool is the Apollonius Theorem which is applicable when a
median is given. Remember, whenever you are asked for squares of sides,
think of the Polonius Theorem, which is 2 times AD squared plus BD square.
The incenter is a point that is equidistant from all three sides of a circle. If the
perpendicular distance from one side is taken and a circle is drawn with that
distance as the length, then all three sides of a triangle will be tangential to
that circle. This circle is inscribed inside the triangle and has an inradius of R
and an incenter of I.
If you draw a line parallel to BC and call it G, and F is parallel to DE, it will
divide triangle JIAE in the same ratio as it divides AD. Thus, we can infer that
the height of the entire triangle is such that H2 is equal to one-third of H1.
The Internal Angle Bisector Theorem may not be used often, but it is a
powerful concept. Congruence and similarity of triangles have wide
applications as they can be used as building blocks for theorems.
For two triangles to be congruent, their shape and size should be equal, and
their areas should also be equal. In right-angle triangles, there is an additional
property called RHS, where if AB is congruent to DE, then you should be able
to pick out the pairs of two sides accordingly.
The SAS property can test whether two triangles are congruent. And what is
our H.S. property?
These are the TR4 checks for congruence of triangles:
, • Checking whether they have the same shape
• Internal angles are the same
• If all three sides and angles are equal, the height of the triangles will
also be equal
This means that even if you have two triangles with different measurements,
but their three angles are equal, then the two triangles will be similar to each
other.
Another test is the "odd tom" or SSS test. Here, if two triangles have:
• Equal corresponding angles
• All three sides in proportion
Then, we can infer that since if two angles are equal, the third angle will have
to be equal because it must add up to 180 degrees. Furthermore, the height of
the larger triangle will be two times the height of the smaller triangle, and the
proportion exists for all parts of this triangle. Essentially, whenever you have
two similar triangles, they will have the same internal angles, corresponding
angles, and all the corresponding sides will be in the same proportion.
Calculating the area of a triangle is usually done by using its height. However,
there are cases where you are only given the three lengths of the triangle, say
ABC. In this scenario, you can calculate the semiperimeter of the triangle which
is half the sum of all three sides. This semiperimeter will be used in the
formula to determine the area of the triangle.
The formula for calculating the area of a triangle given the semiperimeter is:
Area of ABC = 1/2 * a * b * sin(c)
The main objective in this scenario is not actually to calculate the area but
rather to find the inradius and circumradius of the triangle. The formula for
coordinate geometry, which discusses the inradius and circumradius in detail,
will be discussed later in this video.
It is worth noting that an equilateral triangle maximizes the area in a way that
allows you to draw the largest possible circle inside that area. For example, in
an odd shaped triangle where there is a lot of wasted area on one side of the
circle, the area might be larger but the circumradius will always be greater
than two times the inradius of the triangle in all other triangles.
Two properties that will help you in determining the length given the angle or
determining the angle given lengths are the sine rule and cosine rule. While
they are not absolute must-knows, they can significantly reduce the
complexity of questions that are difficult.
As the angle grows, the length of the side of a triangle corresponding to that
angle increases proportionally. The proportionality theorem states that
suppose there is a triangle ABC, and you draw a line DE parallel to BC, then DE
will split AB in the same ratio as it splits AC. This ratio is the ratio of the length
of the part of AD to DB, and AE to EC.
One way to solve questions on triangles is by using the basic theorem where
one side of a smaller triangle is parallel to one side of a bigger triangle. By
using this, the ratios of the areas and lengths of the sides can be figured out.
Another useful tool is the Apollonius Theorem which is applicable when a
median is given. Remember, whenever you are asked for squares of sides,
think of the Polonius Theorem, which is 2 times AD squared plus BD square.
The incenter is a point that is equidistant from all three sides of a circle. If the
perpendicular distance from one side is taken and a circle is drawn with that
distance as the length, then all three sides of a triangle will be tangential to
that circle. This circle is inscribed inside the triangle and has an inradius of R
and an incenter of I.
If you draw a line parallel to BC and call it G, and F is parallel to DE, it will
divide triangle JIAE in the same ratio as it divides AD. Thus, we can infer that
the height of the entire triangle is such that H2 is equal to one-third of H1.
The Internal Angle Bisector Theorem may not be used often, but it is a
powerful concept. Congruence and similarity of triangles have wide
applications as they can be used as building blocks for theorems.
For two triangles to be congruent, their shape and size should be equal, and
their areas should also be equal. In right-angle triangles, there is an additional
property called RHS, where if AB is congruent to DE, then you should be able
to pick out the pairs of two sides accordingly.
The SAS property can test whether two triangles are congruent. And what is
our H.S. property?
These are the TR4 checks for congruence of triangles:
, • Checking whether they have the same shape
• Internal angles are the same
• If all three sides and angles are equal, the height of the triangles will
also be equal
This means that even if you have two triangles with different measurements,
but their three angles are equal, then the two triangles will be similar to each
other.
Another test is the "odd tom" or SSS test. Here, if two triangles have:
• Equal corresponding angles
• All three sides in proportion
Then, we can infer that since if two angles are equal, the third angle will have
to be equal because it must add up to 180 degrees. Furthermore, the height of
the larger triangle will be two times the height of the smaller triangle, and the
proportion exists for all parts of this triangle. Essentially, whenever you have
two similar triangles, they will have the same internal angles, corresponding
angles, and all the corresponding sides will be in the same proportion.
Calculating the area of a triangle is usually done by using its height. However,
there are cases where you are only given the three lengths of the triangle, say
ABC. In this scenario, you can calculate the semiperimeter of the triangle which
is half the sum of all three sides. This semiperimeter will be used in the
formula to determine the area of the triangle.
The formula for calculating the area of a triangle given the semiperimeter is:
Area of ABC = 1/2 * a * b * sin(c)
The main objective in this scenario is not actually to calculate the area but
rather to find the inradius and circumradius of the triangle. The formula for
coordinate geometry, which discusses the inradius and circumradius in detail,
will be discussed later in this video.
It is worth noting that an equilateral triangle maximizes the area in a way that
allows you to draw the largest possible circle inside that area. For example, in
an odd shaped triangle where there is a lot of wasted area on one side of the
circle, the area might be larger but the circumradius will always be greater
than two times the inradius of the triangle in all other triangles.
