Rectangle DEFG is inscribed in equilateral triangle as shown above. What is the area of triangle ABC?
(1) The area of rectangle DEFG is 8√3 square units
(2) The area of triangle AGD is 2√3 square units
Step 1 & 2: Understand Question and Draw Inference
Given
Thus, in order to find the area of the triangle, we need to know the value of x and y.
Step 3 : Analyze Statement 1 independent
Step 4 : Analyze Statement 2 independent
Step 5: Analyze Both Statements Together (if needed)
Therefore, the correct answer is Option C
A frame of uniform width is placed around a square photograph PQRS. If PQ = 10 inches and the area of the frame is 125 square
inches, what is the width of the frame, in inches?
Given:
To find: w = ?
Approach:
1. To find w, we first need to develop a relation in w
2. (Area of frame) = (Area of outer square ABCD) – (Area of inner square PQRS)
Working Out
(Area of frame) = (Area of outer square ABCD) – (Area of inner square PQRS)
At this point, a way to split the middle term 40w into factors doesn’t immediately suggest itself to us. Therefore, we will solve the equation by the method of ‘completing the square’ instead. In this method, we will try to express the quadratic expression in the left hand side of the equation in the form a^{2} +2ab + b^{2} , so that we can then write it as (a+b) . Look below.
Answer: Option B
The rectangle ABCD has its side lengths equal to x and y respectively, where x and y are prime numbers greater than 2. Which of the following cannot be equal to the sum of all the sides of the rectangle ABCD?
I. 64
II. 82
III. 146
Given
To Find
Approach and Working
1. OptionI: 64 → Is divisible by 4, can be equal to 2(x+y)
2. OptionII: 82 → Is not divisible by 4, can never be equal to 2(x+y)
3. OptionIII: 146 → Is not divisible by 4, can never be equal to 2(x+y)
Thus, options II and III can never be equal to the value of 2(x+y).
Hence the correct answer is OPTION E
A wire was initially used to fence a righttriangular plot of land along its perimeter. The lengths, in metres, of the perpendicular sides of the plot were both integers and in the ratio 3:4. The wire was later removed from the triangular plot and used to fence a square plot along its perimeter. When the square plot had been completely fenced, 4 metres of wire still remained. Which of the following could represent the greatest distance between two points in the square plot?
Given
To Find: Which of the 3 values could be the greatest distance between two points in the square plot?
Approach:
Working out
4. We will now check the 3 given possible lengths of the diagonal to see if they lead to integral values of 3x and 4x
Looking at the answer choices, we see that the correct answer is Option B
A student initially draws a rectangle ABCF. Later, he draws an adjacent trapezium FCDE in which side ED is parallel to side FC and drops perpendiculars EG and DH on FC. The dimensions of each side of the polygon ABCDEF, in centimeters, are shown on the figure above. What is the value of x?
(1) The area of the polygon ABCHDEG is 332 square centimeters
(2) If sides AE and BD are joined, the area of the quadrilateral ABDE will be 266 square centimeters
Step 1 & 2: Understand Question and Draw Inference
Given
To find: x = ?
Step 3 : Analyze Statement 1 independent
Step 4 : Analyze Statement 2 independent
So, Statement 2 is sufficient to determine a unique value of x.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at unique answers in each of Steps 3 and 4, this
step is not required
Answer: Option D
In the figure above, a regular hexagon EFGHIJ is inscribed in the rectangle ABCD. If the ratio of the magnitudes of the area and the perimeter of the hexagon is √3/2 , what is the ratio of the magnitudes of the area and the perimeter of the rectangle ABCD?
Given
Approach
Working out
b. As it is a 30^{o}60^{o}90^{o }triangle, with the hypotenuse being equal to x
3. All the 3 other triangles will have the same side lengths and angles as triangle GCH.
4. So, we can write length of the rectangle, l = CD = CH + HI + ID =
In the figure given above, ABCD is a rectangle that is divided into five triangular regions. If triangles I, II and III have equal area and triangles IV and V have equal area, what is the area of rectangle ABCD?
(1) The area of triangle I is 30
(2) The area of triangle V is 15
Step 1 & 2: Understand Question and Draw Inference
To Find: Area of rectangle ABCD = 2x * h
Step 3 : Analyze Statement 1 independent
(1) The area of triangle I is 30
As we know the value of x*h, we can find the value of area of rectangle ABCD.
Sufficient to answer
Step 4 : Analyze Statement 2 independent
(2) The area of triangle V is 15
As we know the value of x*h, we can find the value of area of rectangle
ABCD.
Sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
Two identical square paintings are pasted on a rectangular wooden display board such that the edges of each painting are parallel to the corresponding edges of the display board and the two paintings are symmetrical about an imaginary line passing through the middle of the display board. The margins between the paintings and between the paintings and the board are as shown in the figure above. If the area of the display board is 66 square units, what is the area of each square painting?
Given
To Find: The area of one square painting
Approach and working:
So, to answer the question, we need to find the value of S.
Finding the value of S
Looking at the answer choices, we see that the correct answer is Option D
A flowerbed of uniform width is developed around a rectangular patch of grass such that the outer length of the flowerbed is twothirds more than the length of the rectangular patch and the outer breadth of the flowerbed is 100 percent greater than the breadth of the rectangular patch. If the area of the flowerbed is 8 square units more than twice the area of the rectangular patch, what is the area of the rectangular patch of grass?
Given:
Representing the given scenario visually.
To find : Area of the patch
Approach:
Looking at the answer choices, we see that the correct answer is Option C
In the xycoordinate plane, polygon APBCDA is formed by placing rectangle ABCD and triangle APB as shown above. If sides AB and AD are parallel to the x and the y axes respectively, what is the area of polygon APBCDA?
(1) The coordinates of points A and C are (6, 8) and (12, 12) respectively
(2) The coordinates of point P are (8, 10)
Step 1 & 2: Understand Question and Draw Inference
Given: Rectangle ABCD and triangle APB in the xyplane
To find: Area of polygon APBCDA
Step 3 : Analyze Statement 1 independent
Statement 1 says that ‘The coordinates of points A and C are (6, 8) and (12, 12) respectively’
Step 4 : Analyze Statement 2 independent
Statement 2 says that ‘The coordinates of point P are (8, 10)’
From this statement, we can find neither the area of triangle APB nor the area of rectangle ABCD
So, this Statement is clearly insufficient.
Step 5: Analyze Both Statements Together (if needed)
So, the 2 statements together are sufficient to answer the question Answer: Option C
The length of a rectangle is 2 units greater than its breadth. If a diagonal of the rectangle measures 10 units, what is the perimeter of the rectangle?
Given
To Find: Perimeter
Approach & Working out
Thus, the correct answer is Option C.
In the figure above, ABCD is a quadrilateral in which sides AB and CD are parallel. What is the area of the quadrilateral?
(1) The distance between sides AB and CD is 4 units
(2) The length of side AD is √17 units
Step 1 & 2: Understand Question and Draw Inference
Given:
To find: Area of quadrilateral ABCD
So, to find the area of trapezium, we need to find the value of H
Step 3 : Analyze Statement 1 independent
Step 4 : Analyze Statement 2 independent
25 + 4 – 4y = 17
3^{2} + H^{2} = 25
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this
step is not required
Therefore, the correct answer is Option D.
If parallelogram ABCD has a perimeter of 12 units and ∠ADC = 120 , what is the area of the parallelogram?
(1) BD = 2√3 units
(2) The distance between sides AB and CD is √3 units
Step 1 & 2: Understand Question and Draw Inference
Given
To Find : Area of the parallelogram
Step 3 : Analyze Statement 1 independent
BD = 2√3 units
Step 4 : Analyze Statement 2 independent
h= √3 units
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Steps 3 and 4 each, this step
is not required
Hence the correct answer is Option D .
The figure above shows the walking path in a park, which is to be covered with tiles. Paths A and path B are semicircular in shape with the radius of the inner boundary of the path being equal to 8 feet and 4 feet respectively. If the cost of laying the tiles is $5 per square feet and the path has a uniform width of 2 feet across the park, which of the following is the closest to the amount it will cost, in dollars, to cover the complete walking path with tiles?
Given
To Find
Approach
Working out
2. Area of rectangular regions
3. Area of square region
5. Total cost of laying tiles = 136 * 5 = $680
Hence the correct answer is OPTION C
Is one of the angles of the quadrilateral ABCD equal to 90 degrees?
(1) ABCD is a parallelogram
(2) One of the interior angles of ABCD is equal to 60 degrees
Step 1 & 2: Understand Question and Draw Inference
Given
To Find
Step 3 : Analyze Statement 1 independent
(1) Statement 1 states that "ABCD is a parallelogram"
Step 4 : Analyze Statement 2 independent
(2) Statement 2 states that "One of the interior angles of ABCD is equal to 60 degrees"
Step 5: Analyze Both Statements Together (if needed)
What is the area of rectangle ABCD?
(1) The length of side AB is 2 units more than the length of side BC.
(2) The length of the diagonal AC is 10 units.
Step 1 & 2: Understand Question and Draw Inference
Given: Rectangle ABCD
To Find: Area of rectangle ABCD
Step 3 : Analyze Statement 1 independent
Statement 1 says that ‘The length of side AB is 2 units more than the length of
side BC'
So, AB = BC + 2
Statement 1 alone is not sufficient to answer the question
Step 4 : Analyze Statement 2 independent
Statement 2 says that 'The length of the diagaonal AC is 10 units'
So, AC = 10
Step 5: Analyze Both Statements Together (if needed)
So, the 2 statements together are sufficient to answer the question
Answer: Option C
In the figure given above, a rectangular photograph is placed in a black frame that has a uniform width of x centimeters between the photograph and the outer boundary of the frame. If the difference between the perimeter of the photograph and of the outer boundary of the frame is 32 centimeters, what is the width in centimeters of the black frame?
Given
To Find: Value of x?
Approach
Working out
Hence, the width of the frame is 4 centimeters
Answer : B
In the given figure ABCD is a rectangle and EF  AD. AEG and FEB are two triangles inscribed in the rectangle ABCD such that the area of triangle AEG is half of the area of triangle FEB. What is the area of rectangle ABCD?
(1) EBCF is a square of side length 5 centimeters.
(2) A line passing through point G and parallel to AB divides the rectangle ABCD into 2 equal halves
Step 1 & 2: Understand Question and Draw Inference
(Since EF  AD and ABCD is a rectangle, So, Angle(FEB) =90° »
AE is perpendicular to EG and EB is perpendicular to EF)
To Find: Area of rectangle ABCD
Step 3 : Analyze Statement 1 independent
Step 4 : Analyze Statement 2 independent
However, we do not know anything about the lengths of the sides of the rectangle. So, we cannot calculate the area of the rectangle ABCD
Step 5: Analyze Both Statements Together (if needed)
In the figure given above, the diagonal AC and side DC of square ABCD are extended to form sides of the isosceles triangle CEF where CE = EF. If the length of side CF is 2√2 and the ratio of areas of the triangle CEF to the square ABCD is 1:8, what is the length of the side of square ABCD?
Given
Approach
Working out
3. So, triangle CEF is an isosceles right angled triangle with its hypotenuse equal to 2√2
Using the 45^{o}  45^{o } 90^{o }triangle property, we have CE= EF = 2
4. Hence, area of triangle CEF = ½ * CE * EF = ½ * 2* 2 = 2 square centimeters
5. Using the relation we have Ar(ABCD) = 16
6. Thus, side of square ABCD = √16 = 4 centimeters
Answer : B
Square ABCD and Trapezium PQRS lie between the parallel lines l and m as shown in the figure above. If the area of the square ABCD and trapezium PQRS is equal to 64 square centimeters each, what is the length in centimeters of PQ?
Given
To Find: Length of PQ?
Approach
Working out
Answer : C
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