Areas of Triangles and Special Quadrilaterals Chapter 6 Section 6.7B Areas of Triangles and Special Quadrilaterals Find formulas for the area of triangles, and special quadrilaterals. Use these formulas to find the area of triangles, and special quadrilaterals.

AREA POSTULATES Postulates

AREA FORMULA OF A TRAPEZOID b1 b2 The Shape is a trapezoid. Label the top base b1 and the bottom base b2.

b1 should now be on the bottom and b2 should now be on the top. AREA FORMULA OF A TRAPEZOID b1 b2 b2 b1 Copy the trapezoid upside down to the right of the given one so that they share a common side b1 should now be on the bottom and b2 should now be on the top.

Draw in a height and label it h. Find its length. AREA FORMULA OF A TRAPEZOID b1 b2 h b2 b1 Notice that the new shape is a parallelogram. Find the length of the bottom base of the parallelogram Bottom Base = b2 + b1 = 14 Draw in a height and label it h. Find its length. h = 5

Area of the parallelogram A = bh = 14(5) = 70 square units AREA FORMULA OF A TRAPEZOID b1 b2 h b2 b1 Find the area of the parallelogram using 14 for the base and 5 for the height. Area of the parallelogram A = bh = 14(5) = 70 square units Find the area of one trapezoid. Area of one trapezoid A = ½(70) = 35 square units

Find the area of the parallelogram if b = b1 + b2 and h = h A = bh AREA FORMULA OF A TRAPEZOID b1 b2 h b2 b1 Find the area of the parallelogram if b = b1 + b2 and h = h A = bh A = (b1 + b2 )h What is the area of one trapezoid

AREA FORMULA OF A TRAPEZOID Theorem b2 b1 h

Notice that the Kite ABCD is composed of two triangles, ABC and ADC AREA FORMULA OF A KITE D A B C X Notice that the Kite ABCD is composed of two triangles, ABC and ADC Since the diagonals of a Kite are , BX is the height of ABC and XD is the height of ADC Both triangles have a base of AC

Find the area of each triangle and add them together. AREA FORMULA OF A KITE B C A X D Find the area of each triangle and add them together.

AREA FORMULA OF A KITE B C A X D Factor out the ½(AC).

According to segment addition, BX + XD = BD AREA FORMULA OF A KITE B C A X D According to segment addition, BX + XD = BD

AREA FORMULA OF A KITE Theorem d1 d2

AREA FORMULA OF A RHOMBUS Theorem d1 d2

The shape is a square A = s2 A = 132 A = 169 square units USE A FORMULA TO FIND THE AREA The shape is a square A = s2 A = 132 A = 169 square units

The shape is a triangle A =½bh A = ½(16)(12) A = 96 square units USE A FORMULA TO FIND THE AREA The shape is a triangle A =½bh A = ½(16)(12) A = 96 square units

The shape is a triangle A =½bh A = ½(12)(10) A = 60 square units USE A FORMULA TO FIND THE AREA The shape is a triangle A =½bh A = ½(12)(10) A = 60 square units

The shape is a parallelogram A =bh A = (15)(11) A = 165 square units USE A FORMULA TO FIND THE AREA The shape is a parallelogram A =bh A = (15)(11) A = 165 square units

The shape is a rhombus A = ½d1d2 A = ½(6)(12) A = 36 square units USE A FORMULA TO FIND THE AREA The shape is a rhombus A = ½d1d2 A = ½(6)(12) A = 36 square units

The shape is a trapezoid USE A FORMULA TO FIND THE AREA The shape is a trapezoid

The shape is a rectangle Need to find the base USE A FORMULA TO FIND THE AREA The shape is a rectangle Need to find the base Use the Pythagorean Theorem b A = bh A = 16(12) A = 192 square units b2 + 122 = 202 b2 + 144 = 400 b2 = 256 b = 16

The shape is a trapezoid USE A FORMULA TO FIND THE AREA The shape is a trapezoid Square units

The shape is a kite A = ½d1d2 A = ½(6)(7) A = 21 square units USE A FORMULA TO FIND THE AREA The shape is a kite A = ½d1d2 A = ½(6)(7) A = 21 square units

Since the diagonals of a rhombus bisect each other, this is 4 USING AREA FORMULAS The Shape is a Rhombus A = ½d1d2 64 = ½(16)d2 64 = 8d2 8 = d2 Pythagorean theorem 82 + 42 = x2 64 + 16 = x2 80 = x2 Since the diagonals of a rhombus bisect each other, this is 4

The Shape is a Trapezoid USING AREA FORMULAS The Shape is a Trapezoid

The Shape is a Trapezoid USING AREA FORMULAS The Shape is a Trapezoid

The unshaded region is a rectangle A = bh = 6(16) = 96 square units USING AREA FORMULAS The unshaded region is a rectangle A = bh = 6(16) = 96 square units The shaded region is composed of two congruent triangles A = ½bh 3 6 3

The big triangle is composed of 4 congruent triangles USING AREA FORMULAS The big triangle is composed of 4 congruent triangles Find the area of one of them A = ½bh A = ½(4)(12) = 24 square units Area Shaded = 24 square units Area Unshaded = 3(24) = 72 square units h = 12 4 4

The whole figure is a rectangle Find the area of the whole thing USING AREA FORMULAS The whole figure is a rectangle Find the area of the whole thing A = bh = 20(8) = 160 square units The two triangles are congruent 4 10 A = ½bh A = ½(4)(10) = 20 square units Area Shaded = 2(20) = 40 u2 Area Unshaded = 160 – 40 = 120 u2 10 4

HW #76 Pg 376-378 11-13, 20-25, 27-31, 35-38, 50-52