Problem 2 Finding Area Using a Right Triangle

What is the area of trapezoid PQRS?

You can draw an altitude that divides the trapezoid into a rectangle and a 30 ° − 60 ° − 90 ° 30 degrees , minus , 60 degrees , minus , 90 degrees  triangle. Since the opposite sides of a rectangle are congruent, the longer base of the trapezoid is divided into segments of lengths 2 m and 5 m.

h = 2 3 longer leg  =  shorter leg  · 3 A = 1 2 h ( b 1 + b 2 ) Use the trapezoid area formula .   = 1 2 ( 2 3 ) ( 7 + 5 ) Substitute  2 3  for  h ,   7  for  b 1 ,  and  5  for  b 2 .   = 12 3 Simplify . table with 4 rows and 3 columns , row1 column 1 , h , column 2 equals 2 square root of 3 , column 3 longerleg . equals . shorterleg . middle dot square root of 3 , row2 column 1 , eh , column 2 equals , 1 half , h open , b sub 1 , plus , b sub 2 , close , column 3 cap usethetrapezoidareaformula . . , row3 column 1 , , column 2 equals , 1 half . open , 2 square root of 3 , close . open , 7 plus 5 , close , column 3 cap substitute . 2 square root of 3 , for , h comma 7 , for , b sub 1 , comma , and , 5 , for , b sub 2 , . , row4 column 1 , , column 2 equals 12 square root of 3 , column 3 cap simplify , . , end table

The area of trapezoid PQRS is 12 3 m 2 . 12 square root of 3 . m squared , .

✓ Got It?

2. Reasoning In Problem 2, suppose h decreases so that m ∠ P = 45 m angle , p equals 45  while angles R and Q and the bases stay the same. What is the area of trapezoid PQRS?

Take note Theorem 10-5: Area of a Rhombus or a Kite

The area of a rhombus or a kite is half the product of the lengths of its diagonals.

A = 1 2 d 1 d 2 eh equals , 1 half , d sub 1 , d sub 2

Problem 3 Finding the Area of a Kite

What is the area of kite KLMN?

Find the lengths of the two diagonals:

K M   = 2 + 5 = 7  m and  L N   = 3 + 3 = 6  m . A = 1 2 d 1 d 2 Use the formula for area of a kite .   = 1 2 ( 7 ) ( 6 ) Substitute  7  for  d 1  and  6  for  d 2 .   = 21 Simplify . table with 2 rows and 1 column , row1 column 1 , k m equals 2 plus 5 equals 7 , mand l n equals 3 plus 3 equals 6 m . , row2 column 1 , table with 3 rows and 3 columns , row1 column 1 , eh , column 2 equals , 1 half , d sub 1 , d sub 2 , column 3 cap usetheformulaforareaofakite . . , row2 column 1 , , column 2 equals , 1 half . open 7 close . open 6 close , column 3 cap substitute . 7 , for , d sub 1 , and , 6 , for , d sub 2 , . , row3 column 1 , , column 2 equals 21 , column 3 cap simplify , . , end table , end table

The area of kite KLMN is 21 m 2 . 21 , m squared , .

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3. What is the area of a kite with diagonals that are 12 in. and 9 in. long?