Geometry Chapter 7 Flashcards
Principle 7: A line parallel to a side of a triangle cuts off a triangle…
similar to the given triangle.
If DE is parallel to BC then triangle ADE is similar to triangle ABC
Addition method
A proportion may be changed into an equivalent proportion by adding terms in each ratio to obtain new first and third terms.
If any three terms of one proportion equal the corresponding three terms of another proportion…
ther remaining terms are equal.
If the product of two numbers equals the product of two other numbers,
either pair may be made the means of a proportion and the other pair may be made the extremes.
if 3x=5y, then x:y=5:3 or y:x=3:5 or 3:y=5:x or 5:x=3:y
the length of the leg opposite the 60 degree angle equals
one half the length of the hypotenuse times the square root of 3.
b=1/2c*squareroot of 3
Principle 6: Two right triangles are similar if an accute angle…
of one is congruent to an acute angle of the other.
three or more parallel lines divide…
any two transversals proportionately.
If AB, EF, and CD are all parallel, then a/b=c/d
Principle 2: Corresponding sides of similar triangles are…
In proportion
if c^2 > a^2 + b^2 where c is the longest side of the triangle
then the triangle is an obtuse triangle.
11^2 > 6^2 + 8^2
hence triangle ABC is an obtuse triangle
Corresponding altitudes of similar triangles have…
the same ratio as any two corresponding medians
if triangle ABC is similar to triangle A’B’C’ then h/h’=m/m’
Perimeters of similar polygons have the same ratio as…
any two corresponding sides.
If quadrilateral I is similar to quadrilateral I’ then 34/17=4/2=6/3=10/5=14/7
Inversion method
A proportion may be changed into an equivalent proportion by inverting each ratio.
Corresponding segments of similar triangles are…
in proportion
Eight Arrangements of Any Proportion: Direction Down
Corresponding sides of similar triangles are…
in proportion
In a right triangle, the length of either leg is the mean proportional between…
the length of the hypotenuse and the length of the projection of that leg on the hypotenuse.
AB/BC=BC/BD and AB/AC=AC/AD
The fourth term of a proportion is the…
fourth proportional to the other three taken in order.
2:3=4:x, x is the fourth proportional to 2,3, and 4.
Principle 1: Corresponding angles of similar triangles…
are congruent
If a tangent and a secant intersect outside a circle…
the tangent is the mean proportional between the secant and its external segment
If PA is a tangent, then AB/AP=AP/AC
a 30degree-60degree-90degree triangle is one half…
an equilateral triangle
a = 1/2c. Consider that c=2; then a=1 and the pythagorean theorem gives
b^2 = c^2 - a^2 = 2^2 - 1^2 = 3 or b=square root of 3
Means of a proportion
The middle terms, that is, its second and third terms.
in a:b=c:d , the means are b and c
A 45-45-90degree triangle is one half…
a square.
c^2 = a^2 + a^2 or c = a square root of 2
the ratio of the sides is a:a:c = 1:1: square root of 2
Similar Polygons
Polygons whose corresponding angles are congruent and whose corresponding sides are in proportion. Similar polygons have the same shape although not necessarily the same size.
Eight Arrangements of Any Proportion
Direction Left
If the two means of a proportion are the same,
either mean is the mean proportional between the first and fourth terms.
9:3=3:1, 3 is the mean proportional between 9 and 1.
Principle 9: The altitude to the hypotenuse of a right triangle divides it into…
two triangles which are similar to the given triangle and to each other.
Triangle ABC is similar to ACD and is also similar to CBD
If c^2 < a^2 + b^2 where c is the longest side of the triangle
then the triangle is an acute triangle
9^2 < 6^2 + 8^2
hence triangle ABC is an acute triangle
If two chords intersect within a circle…
the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other
AE X EB = CE X ED
The length of the hypotenuse equals…
the length of a side times the square root of 2
c=a X square root of 2
The length of the leg opposite the 60 degree angle equals the length of
the leg opposite the 30 degree angle times the square root of 3
b = a X square root 3
Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.
C^2 = a^2 + b^2
Subtraction Method
A proportion may be changed into an equivalent proportion by subtracting terms in each ratio to obtain new first and third terms.
If C^2 does not equal a^2 + b^2…
then the triangle is not a right triangle
Principle 4: Two triangles are similar if an angle of one triangle is congruent to…
an angle of the other and the sides including these angles are in proportion
Principle 5: Two triangles are similar if their corresponding sides…
are in proportion
if a/a’=b/b’=c/c’ then triangle ABC is similar to triangle A’B’C’
If two segments are divided proportionately…
1: the corresponding new segments are in proportion
2: the two original segments and either pair of corresponding new segments are in proportion.
If AB and AC are divided proportionately by DE, we may write a proportion such as a/b=c/d using the four segments; or we may write a proportion such as a/AB=c/AC using the two original segments and two of their new segments.
If a line is parallel to one side of a triangle…
then it divides the other two sides proportionately
If DE is parallel to BC, then a/b=c/d
Eight Arrangements of Any Proportion
Direction Right
In any proportion, the product of the means equals…
the product of the extremes.
if a:b=c:d, then ad=bc
Eight Arrangements of Any Proportion: Direction Up
If two secants intersect outside a circle…
the product of the lengths of one of the secants and its external segment equals the product of the lengths of the other secant and its external segment.
AB X AD = AC X AE
Principle 11: Triangles are similar if their sides are respectively…
Perpedicular to each other.
Triangle ABC is similar to triangle A’B’C’
Alternation Method
A proportion may be changed into an equivalent proportion by interchanging the means or by interchanging the extremes.
If a line divides two sides of a triangle proportionately…
it is parallel to the third side.
if a/b=c/d then DE is parallel to BC
Ratio of similitude of two similar polygons
the ratio of any pair of corresponding lines.
Principle 10: Triangles are similar if their sides are respectively…
Parallel to each other
ABC is similar to triangle A’B’C’
triangles similar to the same triangle are…
similar to each other
The length of the altitude of an equilateral triangle equals…
one half the length of a side times the square root of 3
h = 1/2s X the square root of 3
The length of a leg opposite a 45 degree angle equals
one half the length of the hypotenuse times the square root of 2
a=1/2c X the square root of 2
In a series of equal ratios, the sum of any of the numerators…
is to the sum of the corresponding denominators as any numerator is to its denominator.
The length of the altitude to the hypotenuse of a right triangle is the…
mean proportional between the lengths of the segments of the hypotenuse.
Corresponding segments of similar polygons are…
in proportion
The extremes of a proportion
Its outside terms, that is , its first and fourth terms.
a:b=c:d the extremes are a and d.
Principle 3: Two triangles are similar if two angles of one triangle…
are congruent respectively to two angles of the other.
IF angle A=angle A’ and angle B=angle B’ then triangle ABC is similar to triangle A’B’C’
Proportion
An equality of two ratios
2:5=4:10 (or 2/5=4/10) are proportions
A bisector of an angle of a triangle divides…
the opposite side into segments which are proportional to the adjacent sides.
If CD bisects angle C, then a/b=c/d
The length of the leg opposite the 30 degree angle…
equals one half the length of the hypotenuse
a= 1/2c
