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