Geometry 15 Jan 2013 Warm up: Check your homework. For EACH PROBLEM: √ if correct. X if incorrect. Work with your group mates to find and correct any errors. Please use a different color. Use the HW rubric on the purple sheet and grade yourself. I will revise if necessary! 

Objective Students will review derivations and apply formulas for area of geometric shapes to solve problems. Students will take notes, work with their group to solve problems, and present work to the class.

Homework due Today 1) 8.2, pg. 430 +: 1 – 15 odds, 19, 20, 22, 29 2) Study and KNOW the area formulas for rectangles, parallelograms, triangles, trapezoid and kites 3) CHOOSE two of the 5 shapes listed above AND explain HOW WE DERIVED THE FORMULAS (you may use a sketch to help!)

and solving basic problems Homework DUE Friday 1) pg. 435: 1, 5, 9 2) pg. 472: 17, 20, 21, 22 3) study for quiz on area formulas for rectangles, parallelograms, triangles, trapezoids, kites and solving basic problems

Area Formulas

Rectangle

What is the area formula? Rectangle What is the area formula?

What is the area formula? Rectangle What is the area formula? bh We can develop this formula by drawing a rectangle on a grid and counting the squares. Multiplying the number of squares in each row by the # of rows is simply “fast counting”!

What is the area formula? Rectangle What is the area formula? bh What other shape has 4 right angles?

What is the area formula? Rectangle What is the area formula? bh Square! What other shape has 4 right angles?

What is the area formula? Rectangle What is the area formula? bh Square! What other shape has 4 right angles? Can we use the same area formula?

What is the area formula? Rectangle What is the area formula? bh Square! What other shape has 4 right angles? Can we use the same area formula? Yes

Practice! 17m Rectangle 10m Square 14cm

Answers Rectangle A = bh A = (10)(17) A =170 m2 Square A = bh A =196 cm2 14cm

So then what happens if we cut a rectangle in half? What shape is made?

So then what happens if we cut a rectangle in half? Triangle So then what happens if we cut a rectangle in half? What shape is made?

So then what happens if we cut a rectangle in half? Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles

Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula?

Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula?

Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles bh So then what happens to the formula?

Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles bh 2 So then what happens to the formula?

Practice! Triangle 14 ft 5 ft

Answers Triangle 14 ft A = ½ bh A = ½ (14)(5) A = 35 ft2 5 ft

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends? What will the area formula be now that it is a rectangle?

Let’s look at a parallelogram. What happens if we slice off the slanted parts on the ends? What will the area formula be now that it is a rectangle? bh

Parallelogram Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

Parallelogram Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

Parallelogram Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

Rhombus The rhombus is just a parallelogram with all equal sides! So it also has bh for an area formula. bh

Practice! 9 in Parallelogram 3 in Rhombus 2.7 cm 4 cm

Answers A = bh A = (9)(3) A = 27 in2 Parallelogram Rhombus A = bh 2.7 cm A = bh A = (4)(2.7) A = 10.8 cm2 4 cm

Let’s try something new with the parallelogram.

Let’s try something new with the parallelogram. Earlier, you saw that you could use two trapezoids to make a parallelogram.

Let’s try something new with the parallelogram. Earlier, you saw that you could use two trapezoids to make a parallelogram. Let’s try to figure out the formula since we now know the area formula for a parallelogram.

Trapezoid

Trapezoid

Trapezoid So we see that we are dividing the parallelogram in half. What will that do to the formula?

Trapezoid So we see that we are dividing the parallelogram in half. What will that do to the formula? bh

Trapezoid So we see that we are dividing the parallelogram in half. What will that do to the formula? bh 2

But now there is a problem. What is wrong with the base? Trapezoid But now there is a problem. What is wrong with the base? bh 2

Trapezoid So we need to account for the split base, by calling the top base, base 1, and the bottom base, base 2. By adding them together, we get the original base from the parallelogram. The heights are the same, so no problem there. bh 2

Trapezoid So we need to account for the split base, by calling the top base, base 1, and the bottom base, base 2. By adding them together, we get the original base from the parallelogram. The heights are the same, so no problem there. base 2 base 1 base 1 base 2 (b1 + b2)h 2

Practice! 3 m Trapezoid 5 m 11 m

Answers Trapezoid A = ½ h (b1 + b2) A = ½ (5)(3 + 11) A = 35 m2 3 m

Don’t forget ORDER OF OPERATIONS Using Properties of Trapezoids When working with a trapezoid, the height may be measured anywhere between the two bases.  Also, beware of "extra" information.  The 35 and 28 are not needed to compute this area. Area of trapezoid = Don’t forget ORDER OF OPERATIONS P E M/D A/S Find the area of this trapezoid. A = ½ * 26 * (20 + 42) A = 806 u2

Using Properties of Trapezoids Example 2 Find the area of a trapezoid with bases of 10 in and 14 in, and a height of 5 in. A = ½ h (b1+ b2) A = ½ (5)(10 + 14) A = ½ (5)(24) A = 60 in2

Using Properties of Trapezoids Midsegment of a Trapezoid – segment that connects the midpoints of the legs of the trapezoid.

Theorem 6.17 Midsegment Theorem for Trapezoids Using Properties of Trapezoids Theorem 6.17 Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases. ½ (b1+b2) is the length of the midsegment!!

Using Properties of Trapezoids Example 4 Find AB, mA, and mC

So there is just one more left! Let’s go back to the triangle. A few weeks ago you learned that by reflecting a triangle, you can make a kite.

So there is just one more left! Kite So there is just one more left! Let’s go back to the triangle. A few weeks ago you learned that by reflecting a triangle, you can make a kite.

Kite Now we have to determine the formula. What is the area of a triangle formula again?

Kite Now we have to determine the formula. What is the area of a triangle formula again? bh 2

Fill in the blank. A kite is made up of ____ triangles. Now we have to determine the formula. What is the area of a triangle formula again? bh 2 Fill in the blank. A kite is made up of ____ triangles.

Kite Now we have to determine the formula. What is the area of a triangle formula again? bh 2 Fill in the blank. A kite is made up of ____ triangles. So it seems we should multiply the formula by 2.

Kite bh bh *2 = 2

Kite bh bh *2 = 2 Now we have a different problem. What is the base and height of a kite? The green line is called the symmetry line (or a diagonal), and the red line is half the other diagonal.

Let’s use kite vocabulary instead to create our formula. Symmetry Line (or a diagonal) *Half the Other Diagonal A = ½ d1 d2

Using Properties of Kites

Practice! Kite 2 ft 10 ft

Answers Kite A = ½ d1d2 A = (2)(10) A = 20 ft2 2 ft 10 ft 2 ft

Area Kite = one-half product of diagonals Using Properties of Kites Area Kite = one-half product of diagonals

Find the area of the Kite. Using Properties of Kites Example 6 ABCD is a Kite. Find the area of the Kite. 2 4 4 E 4 A = ½ d1d2 A = ½ (4 + 4)(2 + 4) A = ½ (8)(6) A = 24 u2

Final Summary Make sure all your formulas are written down! bh bh (b1 + b2)h 2 2 ½ d1* d2

practice- do 8.1 handout Do all problems. Show work that explains your thinking, clearly and concisely (to the point), and easy for SOMEONE ELSE to follow!  Finished? Work on your homework.

debrief how can you relate each shape to a rectangle? how did you find the formula for area of a triangle? trapezoid? kite? what will you do to help yourself remember the formulas?