5-Minute Check on Lesson 11-1 Transparency 11-2 Click the mouse button or press the Space Bar to display the answers. Find the area and the perimeter of each parallelogram. Round to the nearest tenth if necessary Find the height and base of this parallelogram if the area is 168 square units 6. Find the area of a parallelogram if the height is 8 cm and the base length is 10.2 cm. Standardized Test Practice: ACB D 28.4 cm²29.2 cm²81.6 cm² cm² A = cm² P = 48 cm h = 12, b = 14 units C A = 204 ft² P = 58 ft A = 39.7 m² P = 25.2 m A = in² P = 58 in 11 cm 13 cm 45° 60° 11 in 18 in x + 2 x 6.3 m 17 ft 12 ft

Lesson 11-2 Areas of Triangles, Trapezoids, and Rhombi

Objectives Find areas of triangles –A = ½ bh Find areas of trapezoids –A = ½ (b 1 + b 2 )h Find areas of rhombi –A = ½ d 1 · d 2 (note: this is the one area formula not on SOL formula sheet)

Vocabulary base – the “horizontal” distance of the figure (bottom side) height – the “vertical” distance of the figure area – the amount of flat space defined by the figure (measured in square units) perimeter – once around the figure

Area of Triangles, Trapezoids & Rhombi Triangle Area A = ½ * b * h = ½ * ST * RW h is height (altitude) b is base ( ┴ to h) Trapezoid Area A = ½* h* (b 1 + b 2 ) = ½ * LN * (JK + LM) h is height (altitude) b 1 and b 2 are bases (JK & LM) (bases are parallel sides) Rhombus Area A = ½ * d 1 * d 2 = ½ * AD * BC d 1 and d 2 are diagonals A B C D R ST h b1b1 b2b2 W h JK LM N d1d1 d2d2

Triangle Area Example R ST W h Find the area of triangle RST 10 45° A = ½ bh = ½ 20(h) = 10h square units (side opposite 45°) h = ½ hyp √2 No hypotenuse! So, area = 10(10) = 100 square units ∆ RSW is right isosceles; so legs are equal! h = 10

Trapezoids Area Example h JK L M N 60° A = ½ (b 1 + b 2 )h = ½ ( )(h) = 16h square units (side opposite 60°) h = ½ hyp √3 h = ½ (14) √3 h = 7 √3 So, area = 16(7√3) ≈ square units 14 Find the area of trapezoid JKLM

Rhombi Area Example A B C D 5 Find the area of rhombus ABCD A = ½ (d 1 · d 2 ) = ½ (2(3) · 2(4)) = ½ (48) = 24 square units What if we try to find the area by adding the 4 triangles together? A = 4 (½ bh) = 2bh A = 2(3)(4) = 2 (12) = 24 square units!!

Example 2-1a Substitution Simplify. The area of the quadrilateral is equal to the sum of the areas of Find the area of quadrilateral ABCD if AC = 35, BF = 18, and DE = 10. Area formula Answer: The area of ABCD is 490 square units.

Example 2-1b Find the area of quadrilateral HIJK if IK = 16, HL = 5 and JM = 9 Answer:

Example 2-4a Use the formula for the area of a rhombus and solve for d 2. Rhombus RSTU has an area of 64 square inches. Find US if RT = 8 inches. Answer:US is 16 inches long.

Example 2-4b Trapezoid DEFG has an area of 120 square feet. Find the height of DEFG. Answer:The height of trapezoid DEFG is 8 feet. Use the formula for the area of a trapezoid and solve for h.

Example 2-4c Answer:6 yd Answer: 27 cm b.Trapezoid QRST has an area of 210 square yards. Find the height of QRST. a. Rhombus ABCD has an area of 81 square centimeters. Find BD if AC = 6 centimeters.

Example 2-5a STAINED GLASS This stained glass window is composed of 8 congruent trapezoidal shapes. The total area of the design is 72 square feet. Each trapezoid has bases of 3 and 6 feet. Find the height of each trapezoid. First, find the area of one trapezoid. From Postulate 11.1, the area of each trapezoid is the same. So, the area of each trapezoid is 72  8 or 9 square feet. Next, use the area formula to find the height of each trapezoid.

Example 2-5a Answer:Each trapezoid has a height of 2 feet. Area of a trapezoid Substitution Add. Multiply. Divide each side by 4.5.

Example 2-5b INTERIOR DESIGN This window hanging is composed of 12 congruent trapezoidal shapes. The total area of the design is 216 square inches. Each trapezoid has bases of 4 and 8 inches. Find the height of each trapezoid. Answer:3 in.

Summary & Homework Summary: –The formula for the area of a triangle can be used to find the areas of many different figures –Congruent figures have equal areas Homework: –pg ; 13-18, 30-34