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solid geometry - Area of a rectangle, a triangle

For COMPETITION
Number of Total Problems: 26.
FOR PRINT ::: (Book)

Problem Num : 21

From : AMC10

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

'
An equiangular octagon has four sides of length 1 and four sides of length $frac{sqrt{2}}{2}$ , arranged so that no two consecutive sides have the same length. What is the area of the octagon?
$mathrm{(A) } frac72qquad mathrm{(B) } frac{7sqrt2}{2}qquad mathrm{(C) } frac{5+4sqrt2}{2}qquad mathrm{(D)...$
'

Category Area of a rectangle, a triangle

Analysis

Solution/Answer
The area of the octagon can be divided up into 5 squares with side $frac{sqrt2}2$ and 4 right triangles, which are half the area of each of the squares.
Therefore, the area of the octagon is equal to the area of $5+4left(frac12 ight)=7$ squares.
The area of each square is $left(frac{sqrt2}2 ight)^2=frac12$ , so the area of 7 squares is $frac72Rightarrowmathrm{(A)}$ .

Answer:

Problem Num : 22

From : AMC10

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

'
In $ABC$ we have $AB = 25$ , $BC = 39$ , and $AC=42$ . Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$ . What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$ ?
$mathrm{(A) } frac{266}{1521}qquad mathrm{(B) } frac{19}{75}qquad mathrm{(C) } frac{1}{3}qquad mathrm{(D) } ...$
'

Category Area of a rectangle, a triangle

Analysis

Solution/Answer
The area of a triangle is $frac{1}{2}bcsin A$ .
Using this formula:
$[ADE]=frac{1}{2}cdot19cdot14cdotsin A = 133sin A$
$[ABC]=frac{1}{2}cdot25cdot42cdotsin A = 525sin A$
Since the area of $BCED$ is equal to the area of $ABC$ minus the area of $ADE$ ,
$[BCED] = 525sin A - 133sin A = 392sin A$ .
Therefore, the desired ratio is $frac{133sin A}{392sin A}=frac{19}{56}Longrightarrow mathrm{(D)}$

Answer:

Problem Num : 23

From : AMC10

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

'
In rectangle $ADEH$ , points $B$ and $C$ trisect $overline{AD}$ , and points $G$ and $F$ trisect $overline{HE}$ . In addition, $AH=AC=2$ . What is the area of quadrilateral $WXYZ$ shown in the figure?
$mathrm{(A) } frac{1}{2}qquadmathrm{(B) } frac{sqrt{2}}{2}qquadmathrm{(C) } frac{sqrt{3}}{2}qquadmathrm{(D) ...$

Contents

1 Problem

2 Solution

2.1 Solution 1

2.2 Solution 2

3 See Also

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Category Area of a rectangle, a triangle

Analysis

Solution/Answer
Solution 1

It is not difficult to see by symmetry that $WXYZ$ is a square. Draw $overline{BZ}$ . Clearly $BZ = frac 12AH = 1$ . Then $riangle BWZ$ is isosceles, and is a $45-45-90 riangle$ . Hence $WZ = frac{1}{sqrt{2}}$ , and $[WXYZ] = left(frac{1}{sqrt{2}} ight)^2 = frac 12 mathrm{(A)}$ .
There are many different similar ways to come to the same conclusion using different 45-45-90 triangles.

Solution 2

Draw the lines as shown above, and count the squares. There are 12, so we have $frac{2cdot 3}{12} = frac 12$ .

Answer:

Problem Num : 24

From : NCTM

Type: Complex

Section:solid geometry

Theme:Length
Adjustment# : 0

Difficulty: 1

Category Area of a rectangle, a triangle

Analysis

Solution/Answer

Problem Num : 25

From : NCTM

Type: Understanding

Section:solid geometry

Theme:Length
Adjustment# : 0

Difficulty: 1

Category Area of a rectangle, a triangle

Analysis

Solution/Answer

Problem Num : 26

From : AMC10B

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

'
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
$extbf{(A)} 52 qquad extbf{(B)} 58 qquad extbf{(C)} 62 qquad extbf{(D)} 68 qquad extbf{(E)} 70$
'

Category Area of a rectangle, a triangle

Analysis

Solution/Answer
Answer:

Array ( [0] => 7821 [1] => 7823 [2] => 7836 [3] => 3016 [4] => 3027 [5] => 8152 ) 261 2 3