Chapter 6 Geometry Similarity and Right Triangles
Converse of a Pythagorean theorem
If a squared + b squared= c squared, then a and b are legs and c is the hypotenuse of a unique right triangle right angle c
How is the triangle acute?
If c^2<a^2+b^2
How is the triangle obtuse?
If c^2>a^2+b^2
If the altitude is drawn to the hypotenuse of a right triangle,
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
Converse of transversal Theorem
If the parallel lines divide the transversals proportionally, then the three parallel lines intersect two transversals.
Scale factor
If two polygons are similar, then the ratio of the lengths of two corresponding sides is called
Converse of angle Bisector proportionality Theorem
Ifa ray divides the opposite side into segments whose lengths are proportional to the lengths f the other two sides, the ray bisects an angle of a triangle.
How do you know if points form a segment or a triangle?
It is a segment if the slopes are the same. It is a triangle if the slopes are different.
Geometric Mean
Its of 2 positive numbers And can look like this 2/GM=GM/8
Enlargement
K>1
Are isosceles triangle and scalene triangle (always, sometimes, never) similar?
Never because a scalene triangle has no congruent angles, but an isosceles triangle has 2 congruent angles.
Are a triangle and quadrilateral (always, sometimes, never) similar?
Never because triangle has 3 sides and quad has 4 sides so they can't be proportional .
SAS Similarity Theorem
One side proportional, one angle congruent, one side proportional
What else does scale factor apply to?
Perimeter
Relationship of proportion to ratio
Proportions sets two farms equal to each other.
Similar triangles are __________ congruent triangles.
Sometimes
Are 2 rectangles (always, sometimes, never) similar?
Sometimes Ex: lengths can be 3,2 and 4, 6 which is proportional or 3,2 and 3,5 not proportional
Are a right triangle and isosceles triangle (always, sometimes, never) similar?
Sometimes Ex: similar) 45, 45, 90 not similar) 34, 90, and 60 and 90, 45,45
Are 2 isosceles triangles (always, sometimes, never) similar
Sometimes Ex: similar) 45, 45, 90 not similar) 80,50,50 40,40, 100
Dilation
A stretch or shrinking that creates a similar figure If the dilation is with respect to origin, (x,y)——->(kx, ky) indicates a scale factor of k
Angle Bisector proportionality Theorem
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to lengths of other two sides.
HL Similarity Theorem
Triangles are right, 2 legs are proportional.
AA Similarity Postulate
2 angles congruent
Pythagorean triple
3 positive integers a,b,c such that a squared + b squared= c squared Ex: 3,4,5
Reduction
0<k<1
SSS Similarity Theorem
All 3 sides are proportional
Congruent triangles are ________ similar triangles.
Always
Are 2 equilateral triangles (always, sometimes,never) similar
Always (equilateral triangles always have degrees of 60)
Are 2 squares (always, sometimes, never) similar?
Always it has all 90 degrees and all sides are equal, so it has to be proportional
In a right isosceles triangle, what is relationship between the two smaller triangles formed when an altitude is drawn to the hypotenuse?
Congruent
Pythagorean theorem
If "a" and "b" are legs of a right triangle and "c" is the length of the hypotenuse then a^2+b^2=c^2
Theorem 6.6 (transversal)
If 3 parallel lines intersect 2 transversales, then they divide the transversals proportionally
Converse of triangle proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side. The "parallel part" of midsegment Theorem is a special case where ratios = 1
Triangle proportionality Theorem
If a line parallel to 1 side of a triangle intersects the other 2 sides, then it divides the two sides proportionally.
How is area related to SF?
The area of a rectangle equals the area of the original rectangle multiplied by the square of the Scale factor.
Geometric mean theoream
The length of the interior altitude of a right triangle is the GM of the lengths if the hypotenuses 2 segments. The length of a leg (altitude) of a right triangle is the GM of the length of the hypotenuse and length of the adjacent segment of the hypotenuse.
Why is the ratio of legs and segments of the base always 1:1 in an ISOSCELES TRIANGLE?
The ratio of the leg lengths is 1:1 because it's an isosceles triangle and the ratio of leg lengths equals the ratio of the base segments.
Similar polygons
They have congruent angles and side lengths are proportional They have the same shape
