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Geometry > Geometry Chapter 7 > Flashcards

Geometry Chapter 7 Flashcards

1
Q

Principle 7: A line parallel to a side of a triangle cuts off a triangle…

A

similar to the given triangle.

If DE is parallel to BC then triangle ADE is similar to triangle ABC

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2
Q

Addition method

A

A proportion may be changed into an equivalent proportion by adding terms in each ratio to obtain new first and third terms.

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2
Q

If any three terms of one proportion equal the corresponding three terms of another proportion…

A

ther remaining terms are equal.

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3
Q

If the product of two numbers equals the product of two other numbers,

A

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

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3
Q

the length of the leg opposite the 60 degree angle equals

A

one half the length of the hypotenuse times the square root of 3.

b=1/2c*squareroot of 3

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4
Q

Principle 6: Two right triangles are similar if an accute angle…

A

of one is congruent to an acute angle of the other.

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5
Q

three or more parallel lines divide…

A

any two transversals proportionately.

If AB, EF, and CD are all parallel, then a/b=c/d

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5
Q

Principle 2: Corresponding sides of similar triangles are…

A

In proportion

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5
Q

if c^2 > a^2 + b^2 where c is the longest side of the triangle

A

then the triangle is an obtuse triangle.

11^2 > 6^2 + 8^2

hence triangle ABC is an obtuse triangle

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6
Q

Corresponding altitudes of similar triangles have…

A

the same ratio as any two corresponding medians

if triangle ABC is similar to triangle A’B’C’ then h/h’=m/m’

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6
Q

Perimeters of similar polygons have the same ratio as…

A

any two corresponding sides.

If quadrilateral I is similar to quadrilateral I’ then 34/17=4/2=6/3=10/5=14/7

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8
Q

Inversion method

A

A proportion may be changed into an equivalent proportion by inverting each ratio.

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9
Q

Corresponding segments of similar triangles are…

A

in proportion

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10
Q

Eight Arrangements of Any Proportion: Direction Down

A
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11
Q

Corresponding sides of similar triangles are…

A

in proportion

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12
Q

In a right triangle, the length of either leg is the mean proportional between…

A

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

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13
Q

The fourth term of a proportion is the…

A

fourth proportional to the other three taken in order.

2:3=4:x, x is the fourth proportional to 2,3, and 4.

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13
Q

Principle 1: Corresponding angles of similar triangles…

A

are congruent

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14
Q

If a tangent and a secant intersect outside a circle…

A

the tangent is the mean proportional between the secant and its external segment

If PA is a tangent, then AB/AP=AP/AC

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14
Q

a 30degree-60degree-90degree triangle is one half…

A

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

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15
Q

Means of a proportion

A

The middle terms, that is, its second and third terms.

in a:b=c:d , the means are b and c

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16
Q

A 45-45-90degree triangle is one half…

A

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

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17
Q

Similar Polygons

A

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.

18
Q

Eight Arrangements of Any Proportion

Direction Left

A
19
Q

If the two means of a proportion are the same,

A

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.

20
Q

Principle 9: The altitude to the hypotenuse of a right triangle divides it into…

A

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

20
Q

If c^2 < a^2 + b^2 where c is the longest side of the triangle

A

then the triangle is an acute triangle

9^2 < 6^2 + 8^2

hence triangle ABC is an acute triangle

21
Q

If two chords intersect within a circle…

A

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

21
Q

The length of the hypotenuse equals…

A

the length of a side times the square root of 2

c=a X square root of 2

22
Q

The length of the leg opposite the 60 degree angle equals the length of

A

the leg opposite the 30 degree angle times the square root of 3

b = a X square root 3

23
Q

Pythagorean Theorem

A

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

24
Q

Subtraction Method

A

A proportion may be changed into an equivalent proportion by subtracting terms in each ratio to obtain new first and third terms.

25
Q

If C^2 does not equal a^2 + b^2…

A

then the triangle is not a right triangle

26
Q

Principle 4: Two triangles are similar if an angle of one triangle is congruent to…

A

an angle of the other and the sides including these angles are in proportion

28
Q

Principle 5: Two triangles are similar if their corresponding sides…

A

are in proportion

if a/a’=b/b’=c/c’ then triangle ABC is similar to triangle A’B’C’

29
Q

If two segments are divided proportionately…

A

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.

31
Q

If a line is parallel to one side of a triangle…

A

then it divides the other two sides proportionately

If DE is parallel to BC, then a/b=c/d

32
Q

Eight Arrangements of Any Proportion

Direction Right

A
34
Q

In any proportion, the product of the means equals…

A

the product of the extremes.

if a:b=c:d, then ad=bc

35
Q

Eight Arrangements of Any Proportion: Direction Up

A
36
Q

If two secants intersect outside a circle…

A

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

37
Q

Principle 11: Triangles are similar if their sides are respectively…

A

Perpedicular to each other.

Triangle ABC is similar to triangle A’B’C’

38
Q

Alternation Method

A

A proportion may be changed into an equivalent proportion by interchanging the means or by interchanging the extremes.

40
Q

If a line divides two sides of a triangle proportionately…

A

it is parallel to the third side.

if a/b=c/d then DE is parallel to BC

42
Q

Ratio of similitude of two similar polygons

A

the ratio of any pair of corresponding lines.

43
Q

Principle 10: Triangles are similar if their sides are respectively…

A

Parallel to each other

ABC is similar to triangle A’B’C’

45
Q

triangles similar to the same triangle are…

A

similar to each other

46
Q

The length of the altitude of an equilateral triangle equals…

A

one half the length of a side times the square root of 3

h = 1/2s X the square root of 3

47
Q

The length of a leg opposite a 45 degree angle equals

A

one half the length of the hypotenuse times the square root of 2

a=1/2c X the square root of 2

49
Q

In a series of equal ratios, the sum of any of the numerators…

A

is to the sum of the corresponding denominators as any numerator is to its denominator.

50
Q

The length of the altitude to the hypotenuse of a right triangle is the…

A

mean proportional between the lengths of the segments of the hypotenuse.

51
Q

Corresponding segments of similar polygons are…

A

in proportion

52
Q

The extremes of a proportion

A

Its outside terms, that is , its first and fourth terms.

a:b=c:d the extremes are a and d.

54
Q

Principle 3: Two triangles are similar if two angles of one triangle…

A

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’

55
Q

Proportion

A

An equality of two ratios

2:5=4:10 (or 2/5=4/10) are proportions

56
Q

A bisector of an angle of a triangle divides…

A

the opposite side into segments which are proportional to the adjacent sides.

If CD bisects angle C, then a/b=c/d

57
Q

The length of the leg opposite the 30 degree angle…

A

equals one half the length of the hypotenuse

a= 1/2c

Geometry (11 decks)

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