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Geometry
Lesson 7-3
Link to Prior Learning
5-Minute Check Transparency 7-3
Objective(s)
The student will learn to:
1. Use properties of 450-450-900 triangles.
2. Use properties of 300-600-900 triangles.
Instructional Input
This section is probably one of the most important sections in this book.
These special triangles are on all standardized tests as well as the EOC.
Just telling the students the theorems will not help them remember them.
There has to be a hands-on way to get them to derive the formulas for
these triangles.
Some time needs to be spent reviewing isosceles triangles and isosceles
right triangles. The students must recognize quickly that the legs are
congruent. Therefore there is only one unknown to be found. You can give
the students different isosceles triangles and let them find the hypotenuse,
make a chart, and compare the answers. Then let them make a conjecture
about a 45-45-90 triangle.
The discovery lesson that I like to do can work with both the 45-45-90
triangle and the 30-60-90 triangle. Give each student in the group a
different 45-45-90 triangle. Have them measure all the sides of the
triangle. Have a chart to fill in the measurements. Then have columns for
ratios of leg: leg and hypotenuse: leg. Now ask them to change these ratios
to decimals. The legs will have ratio of 1 since they are congruent. They will
get approximately 1.414 for the ratio of the hypotenuse to a leg. Then using
their calculator have them find the 2 . Therefore given the leg of an
isosceles right triangle they multiply by 2 to get the hypotenuse. Also
remember to ask them how to find a leg given the hypotenuse. You will
probably have to review how to divide by a square root.
This is a good time to tell them they should know how to give the answer as a
square root and also as a decimal.
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To prove the 30-60-90, give each member of the group a different
equilateral triangle. Take a few minutes to review all the information they
should know (equal sides, equal angles, 60 angles). Using another sheet of
paper (so they will have a right angle at the corner of the paper) have them
draw an altitude to one side of the triangle. (I would make all my sides even
numbers just to get the easy answers.)
Have a data chart to fill in as the measurements are made. Ask them what
kind of new triangles are formed? What are the measurements of the
angles of the new triangles? The altitude is opposite which angle? Using the
smaller right triangle that is formed, the altitude is which leg, longest or
shortest? Why is it the longest? The shortest is opposite which angle?
Now measure the sides of the small right triangle. It is a 30-60-90
triangle. Have columns for the ratio of longest leg: shortest leg and
hypotenuse: shortest leg. What is the relationship between the shortest
leg and the hypotenuse? The hypotenuse is twice the shorter leg. What is
the ratio of the longer leg to the shorter leg? The students should get
approximately 1.732. Using a calculator have them find the 3 . Therefore
the longer leg is equal to the shorter leg multiplied by the 3. Now you
need to work examples with any of the sides missing and have them find the
others. Again, you will probably have to review how to divide by square
roots.
This may seem like a waste of time to prove these theorems. Anything you
can do that will make the students remember the theorems is not a waste of
time. Always have the students write in their own words what they
discovered. It does not matter if it is not exactly what is in the book as
long as it is correct.
Example 4 is another good example to practice coordinate geometry. You
might want to ask them if they could tell if a triangle is a right triangle given
the coordinates. If they cannot graph it remind them they can do the slopes
and if they are negative reciprocals then it would be perpendicular lines and
form a right angle.
Have them work examples where they have to do coordinate geometry and
find the missing coordinate.
Word Wall Words
No new words.
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Independent Practice or Homework
Pages 361-362, 12-18, 21, 22
Closure
Practice Quiz #1 page 363.