Similarity - Class Practice
midsegment of a triangle
A segment that joins the midpoints of two sides of the triangle
What is the difference between congruent figures and similar figures.
Congruent figures are the exact same size and shape, but similar figures are the same shape but not necessarily the same size.
What do we know about the corresponding ANGLES of similar figures?
Corresponding angles of similar figures are CONGRUENT.
What do we know about the corresponding SIDES of similar figures?
Corresponding sides of similar figures are PROPORTIONAL.
Similar Polygons
Figures that are the same shape but different sizes because all of their corresponding angles are congruent and all of their corresponding sides are proportional. These figures also have perimeters that are proportional.
ratio
The quantitative relation between two amounts showing the number of times one value contains or is contained within the other.
Leg Mean
a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude
Similar polygons
Same shape and angles; different sizes
___________ are two figures that have the same shape but not necessarily the same size.
Similar Figures
How do you use proportions to find a missing side length in similar figures?
Set up an equation as shown. Solve for x by cross multiplying. You will get the equation 3x = 6. Divide both sides by 3 to get the final answer: x = 2.
Mean Proportion
proportion in which the means are equal
Geometric Mean
Number that when substituted for X will make a porportion true
Find the Scale Factor: 10 cm = 5 m
1 cm = 50 cm
Similarity Statement
A statement that matches the corresponding angles and sides from one figure to another.
right isosceles triangle
A triangle where two sides match, while one does not. Also one angle is 90 degrees
Geometric mean
For two positive numbers a and b, the positive number x such that a/x = x/b
Means
Middle terms of a proportion
Dilation
Stretches or shrinks
Extremes
The first and last terms of a proportion
When should we use indirect measurement?
When we need to find measurements that are too difficult to measure directly.
similar figures
figures that have the same shape but not necessarily the same size
AA Similarity
two pairs of angles of triangles are congruent = similar
A skate park is 24 yards wide by 48 yards long. If we used scale of 1 inch = 32 yards, what is the width and length of the scale drawing?
0.75 yards wide by 1.5 yards long
An area rug is 9 feet wide. In a photograph, the image of the rug is 3 inches wide. What is the scale?
1 inch = 3 feet
proportion
1. The relationship of one thing to another in size, amount, etc. 2. Size or weight relationships among structures or among elements in a single structure.
An architect is using a scale of 1 in. = 10 ft. What is this scale as a fraction?
10 ft. /1 in.
A five foot parking meter casts a 6 foot long shadow. At the same time of day, how long is the shadow of a nearby 80 for building?
96 foot long shadow
Extended Ratio
A comparison of three or more quantities written as a:b:c or a to b to c.
Ratio
A comparison of two or more quantities written as a/b, a:b, or a to b.
Scale Drawings
A drawing that shows a real object with accurate proportions reduced or enlarged by a certain amount.
triangle midsegment theorem
A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side
Scale factor
A number that multiplies
Ratio
A quotient a/b if a and b are two quantities that are measurd in the same units
Enlargement
Act of making something bigger
Reduction
Act of making something smaller
Proportion
An equation stating that two ratios are equal.
What is a proportion?
An equation stating that two ratios are equal.
Proportion
An equation that equates two ratios
Triangle ABC is similar to triangle DEF. If the measure of angle A is 52 degrees and the measure of angle E is 67 degrees, what is the measure of angle C?
Angles A and D are both 52 degrees. Angles B and E are both 67 degrees. We can find the measures of both angles C and F by adding 52 + 67 = 119. Subtract 180 - 119 = 61 degrees.
In order for two figures to be similar figures what two things have to be true about the two figures?
Corresponding angles are congruent and corresponding sides are proportional.
Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional length.
triangle proportionality theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
Side Side Side Similarity (SSS)
If all three pairs of corresponding sides are proportional, then the triangles are similar.
SAS Similarity
If an angle of 1 triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
SSS Similarity
If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar
Proportional Parts and Parallel Lines
If three or more parallel lines intersect two transversals, then they cut the transversals proportionally.
Angle Angle Similarity (AA)
If two pairs of corresponding angles are congruent, then the triangles are similar.
Side Angle Side Similarity (SAS)
If two pairs of corresponding sides are proportional AND their included angles are congruent, then the triangles are similar.
Converse of the Triangle Proportionality Theorem
If two sides of a triangle are segments of proportional length, then the line that divides them is parallel to the third side of the triangle.
Triangle LMN is similar to triangle RST. What is the value of LN if RT is 9 inches, MN is 21 inches, and ST is 7 inches.
LN = 27 inches
__________ is a ratio of a given length on a scale drawing or model to the corresponding length on the actual object.
Scale
What is used to represent an object that is too large or too small to be drawn or built at actual size?
Scale drawing or scale model
When talking about a scale drawing or model, _____________ is determined by the ratio of a given length on the drawing or model to its corresponding length on the actual object. How do we write this as a fraction?
Scale: (Actual measure) / (Drawing Measure)
Similar Polygons
Similar polygons have the exact same shape but not necessarily the same size. The corresponding angles are congruent and the side lengths are proportional.
Suppose A Monument in Texas casts a shadow of 285 feet. At the same time, a nearby tourist, who is 5 feet tall casts a 2.5 foot shadow. How tall is the Monument?
The Monument is 570 feet tall.
Scale Factor
The ratio of any two corresponding lengths in two similar geometric figures.
Scale Factor
The ratio of corresponding sides. Order matters.
The height of an amusement park ride is 157.5 feet. If the ride's shadow is 60 feet long, how long will a person's shadow be if the person is 5.3 feet tall?
The shadow will be about 2 feet long.
Three similar triangles
if 1 triangle can be transformed into another and the result is equal to a 3rd triangle the first and 3rd are similar
Altitude Mean
that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse.
