Similar Figures & Scale Drawings 6-3
Proportions are used a lot in Geometry
Two figures are similar if they are proportional
(1) they are exactly the same shape but different sizes
(2) each corresponding side of one of the figures is the same ratio of every other corresponding side.
(3) every corresponding angle is exactly the same measurement ( congruent)
All circles and all squares are similar.
This is actually true for every regular polygon.
What is a regular polygon? All sides of the polygon are equal and all angles are equal!! Or.. a polygon with congruent sides and congruent angles. For example:
Equilateral triangle
Square
A good example of a regular octagon would be a stop sign.
When you name similar figures, you should go in the same order of their corresponding parts.
The symbol for SIMILAR looks like 1 of the 2 wavy lines that means approximately equal The symbol ~ means “is similar to”
SCALE DRAWING PROBLEMS ARE PERFECT TO SOLVE WITH PROPORTIONS!
For example a map says that 3 cm: 1000 miles
You measure that you need to go 5.4 cm on the map
How many miles is that?
Set up a proportion with the labels
cm/mi = cm/mi
3/100 = 5.4/x
Use cross products to solve
3x = 1000(5.4) but don’t multiply out yet
divide both sides by 3 so
x = 1000(5.4)/3 use your divisibility rules to simplify first
x = 1000(1.8) because 5.4/3 = 1.8
x = 1800 miles. Make sure your answer seems reasonable. That is, you know if 3cm is 1000 miles , then 6 cm would represent 2000 miles so your answer should be less than 2000 miles.
Indirect measurement
Person 5 ft tall casts a 4 ft shadow. A tree casts a 10 foot shadow. How tall is the tree. Set up proportions but make sure your labels match
5/4 = x/ 10 5(10) = 4x again don’t multiply out first
divide both sides by 4 so 5(10)/4 = x 25/2 = x so the tree must be 12.5 ft tall
set up
actual height of person/shadow = actual height of tree/shadow
person shadow/ tree shadow = person’s height/ tree’s height
You will arrive at the same answer!!
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