Topic 12 Right Triangles

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Topic 12 - Right Triangles
And the PYTHAGOREAN Theorem
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Topic 9Right Triangles

• Radicals Square Roots
• Pythagorean Theorem
• - How does this work??
• Converse of the Pythagorean Theorem
• Remember Use with obtuse and acute triangles
• Applications

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Day 1 - Radicals
- Altitude
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Radicals Square Roots
Make a table of perfect squares.
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• Between what two perfect squares do the following
  values fall between? EX 6.59 falls between 36
  and 49.

a. 5.62
b. 15.63
c. 27.92
d. 69.75
e. 112.53
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Multiply Radicals
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More radicals!
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Cabri Jr. TI-83Plus

• Go to APPS, then to Cabri Jr.
• Press any key
• You should see an arrow on the screen. If there
  is anything else, press Graph , then arrow down
  to Clear, then arrow over to All.

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Cabri Jr. (continued) TI-83Plus

• EXPLORE Press F1, F2, F3, F4, F5 keys to see
  what is there to use.
• Clear ALL
• F2 Triangle Form a right triangle with legs of
  different lengths. Press ENTER to begin drawing,
  and ENTER at the right angle, then ENTER again at
  the third vertex.
• Compare with your neighbor!

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Cabri Jr. (continued) TI-83Plus

• F5 measure. D Length ENTER.
• Move cursor to any side of the triangle so a bold
  arrow appears. ONLY ONE SIDE of the triangle
  should be scrolling. Move cursor until this
  happens that is the side that will be measured.
• Press Enter, then arrow to move the measurement
  hand until you can see it on the segment. Press
  ENTER when you are happy with its location.
• Repeat with the other 2 sides.
• Write these measurements on your paper.

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Cabri Jr. (continued) TI-83Plus

• Go to the home screen of your calculator to prove
  that the triangle you drew is truly a right
  triangle.

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Cabri Jr. (continued) TI-83Plus

• To go back to Cabri Jr., Go to APPS, then Cabri
  Jr. and your diagram will be there!
• To get rid of the diagram, F5, Clear, All.

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DAY 2
- Right Triangle Squares (What's
that?) - Algebraic representation - Coordinate
representation
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Cabri Jr. (continued) TI-83Plus

• New Diagram!!
• Draw a segment, put your cursor on the segment so
  it is scrolling, select perpendicular from the
  menu, and ENTER.
• Use the cursor to move the perpendicular line to
  one of the endpoints. ENTER.
• Highlight the perpendicular line. Go to POINT,
  and point ON to put a point on the perpendicular
  line.
• Draw a SEGMENT to form a right triangle.

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Cabri Jr. (continued) TI-83Plus

• Measure each side and prove that this is a right
  triangle.
• Challenge
• Draw a segment, find its midpoint, draw a
  perpendicular line through the midpoint, forming
  2 congruent right triangles.
• Why are they congruent?
• Compare their areas perimeters.

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• Pyth Application http//www.classzone.com/books/g
  eometry_2007_tx/page_build.cfm?idlesson1ch7
• Example 1

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Find the missing side.
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• Page 462 Investigation
• http//davinci.keypress.com/KeyPressPortalV2.0/Vie
  wer/HomeFrame.htm

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• Page 464 Example A and Example B
• http//davinci.keypress.com/KeyPressPortalV2.0/Vie
  wer/HomeFrame.htm

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• Find the perimeter of the triangle with the
  lengths of the
• legs (x 6) and (x -3) in terms of x.

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Draw the indicated triangle and label the sides
with the lengths.

• ?LMO with segment LO as the hypotenuse.
• Find the value of x and the lengths of each side.
• Find the perimeter in terms of x.
• Find the area in terms of x.

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Find the value of x.
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Find the length of the missing side of the
base.Find the area of the base.Find the volume
of the container if it is 1 unit tall.Find the
volume of the container if it is 2 units
tall.Find the volume of the container if it is 3
units tall.Find the volume of the container if
it is 2 units tall.
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Using the previous diagram

• If the volume of the prism is 300 cm³, how tall
  is it?
• If the volume of the prism is 3000 cm³, how tall
  is it?

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Using the previous diagram
If the area of the largest rectangular side of
the right triangular prism is 78 cm², what is
the volume?
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• Page 465 Various problems
• http//davinci.keypress.com/KeyPressPortalV2.0/Vie
  wer/HomeFrame.htm

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Now for some right triangles using coordinates.

• Problem 1
• Graph the points (1, 1 and (1, 5)
• ?Name one point that would form a right
  isosceles triangle with these given points.
• ?Name another point that would form a
  right isosceles triangle that is a reflection of
  the first.
• more

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• ?Name two more points that would reflect both of
  those triangles. Are the new triangles right
  isosceles triangles?
• ?Name a point that would form a right isosceles
  triangle with these given points but the right
  angle is not located at either of the two given
  points.
• What did you have to know to do this?

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Graph 3 points that form a right scalene
triangle, with no horizontal or vertical sides.
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DAY 3
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i) Find the length of the side labeled
x.ii) Find the area of the rectangle.iii) If
the lengths of the sides were doubled,
what would be the new area?
Day 3
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Pythagorean Triples

• Two sets of triples are
• 3, 4, 5
• and
• 5, 12, 13
• What makes these different from
• 4, 6, 7.211

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Triple Families

• Name the first three sets of
• triple families
• 3, 4, 5
• 5, 12, 13
• 8, 15,17
• 7, 24, 25
• 11, 60, 61

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Cabri Jr. ActivityForming a perpendicular line

• http//www.cabrijr.com/viewactivity.php?actionvie
  wactivityactivityid40

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• What is the range of measures that TI will fall
  between?
• Find the missing sides and area and perimeter.

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The triangle below is isosceles.

• Find x.
• 7
• b. 23
• c.
• d. 5
• e.

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• Determine whether the following is a right
  triangle or not a right triangle.
• A. 4, 5, 3
• B. 16, 30, 34
• C. 39, 10, 36
• What are the base triples?

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• A 15 foot ladder is leaning against a house. It
  touches the ground 9 feet from the house. How
  far above the ground does it touch the house?

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Pythagorean Theorem Applications
Height of the roller coaster is 60
feet. Horizontal distance is 80 feet. What is
the distance traveled by the coaster?
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Remember!!!
Add this to your notes or index card if
needed!!
Geometers Sketchpad Pyth oblique
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Converse of the Pythagorean Theorem

• The given segment lengths of 6, 7, and 11 units
  form
• a. no triangle
• b. an acute triangle
• c. an obtuse triangle
• d. a right triangle

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• The given segments lengths of 1 unit, 3.6 units,
  and 3 units form
• a. no triangle
• b. an acute triangle
• c. an obtuse triangle
• d. a right triangle

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• Page 468
• http//davinci.keypress.com/KeyPressPortalV2.0/Vie
  wer/HomeFrame.htm

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• Page 469 Historical connection
• http//davinci.keypress.com/KeyPressPortalV2.0/Vie
  wer/HomeFrame.htm

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Day 4 Activity
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• Pythagorean Activity
• http//www.keymath.com/x3341.xml

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• Pyth Application http//www.classzone.com/books/g
  eometry_2007_tx/page_build.cfm?idlesson1ch7

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Day 5 Applications Area Perimeter
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Applications
Find the length of a diagonal of a square whose
perimeter is 80 cm.
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• Pyth Application http//www.classzone.com/books/g
  eometry_2007_tx/page_build.cfm?idlesson1ch7

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• Find the altitude of an equilateral triangle if
  each side is 30 units long.

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Measure angles in right triangles

• What are the relationships of the angle measures.
  
• In a right triangle, what is the relationship of
  the angles.
• - If both angles are congruent, what type of
  triangle is formed?
• - As one acute angle gets larger, what is
  the effect on the other acute angle and the
  lengths of the sides?

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Remember acute angles in a right triangle are
_______
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Discovering Geometry

• Page 207 7
• http//davinci.keypress.com/KeyPressPortalV2.0/V
  iewer/HomeFrame.htm

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Applications

• Discovering Geometry textbook

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Coordinate

• On graph paper, find three points that form a
  right triangle.
• Given the points, (-3, 6) and (3, 2) find a third
  point that will form a right triangle.
• Do the points (-2, 2), (3, 0), and (-4, 3) form a
  right triangle?

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Review

• Do the following lengths form a right triangle?

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Somebodys moving!
A wooden packing crate has been built to contain
the contents of Joes bedroom during his move.
He has clothing and the usual stuff, and also
wants his mattress to be put inside the crate for
safe-keeping. But when they try to put it in, it
will not fit against one of the sides of the
crate. The bottom of the rectangular prism
shaped crate has the dimensions 60 in x 42 in.

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• What is the maximum length of the mattress (to
  the nearest inch) of the mattress ---
• if it fits??