Problem Solving - 48

What is the angle formed by the diagonals of two adjacent surfaces of a cube?

A) 45
B) 60
C) 90
D) 30
E) None of these


Answer: B

Draw from two adjacent surfaces of a cube the diagonals.

Now see that it forms an equilateral triangle if we draw a third diagonal i.e diagonal in green color.
Thus angles of the triangle will be 60 degrees; hence the answer.

Data Sufficiency - 44

In triangle ABC, AB has a length of 10 and D is the midpoint of AB. What is the length of line segment DC?

(1) Angle C= 90
(2) Angle B= 45

Answer: A

From statement (1): it is given that angle C = 90 degrees ...this implies that ABC is a right angle triangle with AB as the hypotenuse and DC as the median. We know that --- In all right triangles, the median on the hypotenuse is the half of the hypotenuse. Hence DC=5

Data Sufficiency - 43

In the figure shown, what is the value of x?

(1) The length of line segment of QR is equal to the length of line segment RS
(2) The length of line segment of ST is equal to the length of line segment TU

Answer: C

From statement (1): Length of line segment of QR is equal to the length of line segment RS ..this implies angle RQS = angle RSQ = p(say)
From statement (2): Length of line segment of ST is equal to the length of line segment TU .. this implies angle TUS = angle TSU = q(say)

Hence p+p+angle QRS = 180 --- eq(1) and q+q+angle UTS = 180 --- eq(2)
Thus, p+q+x = 180

Now because angle RPT = 90, QRS+UTS= 90
adding eq(1) and eq(2) we get:
2p+2q+QRS+UTS = 360
2p+2q+90=360
p+q = 270/2 = 135

Now x = 180-p-q..hence the answer C
x = 180 - (p+q) = 180 - 135 = 45

Data Sufficiency - 41

Sania has a circular garden in her backyard. She puts poles A,B and C on the circumference of her garden. Then she ties ropes between these poles. Is length of one of the ropes is equal to the diameter of her garden?

1. Slope of line joining pole A and B is 3/4 and slope of line joining poles B and C is -4/3
2. Length of line joining pole A and B is 12 and length of line joining B and C is 5


Answer: A

From statement (1): Given that the slope of line AB is 3/4 and slope of line BC is -4/3. This implies that the product of slopes = -1. Hence AB perpendicular BC and B is a right angle. Thus AC is a diameter which implies ABC form a semi-circle.
Hence sufficient

From statement (2): Given that length of line AB is 12 and length of BC is 5. However this does not imply that ABC is a right angled triangle. We can draw number of different triangles with the same given two sides but with different third side.
Hence insufficient

DS Question - 27

The area of a parallelogram is 100. What is the perimeter of the parallelogram ?

1) the base of the parallelogram is 10
2) one of the angles of the parallelogram is 45 degree

Answer: C

From (1) - For a parallelogram, Area = Base*Height => Height = 10 - There are infinite ways to draw a parallelogram with 100 as area, as long as the height is 10 units.( parallelograms with varying slants from 1 to 179 degrees) -- insufficent

From (2) - one of the angles is 45 degrees - hence the opposite angle is 45 degrees too, and the two remaining angles wil be 135 degrees each. But this doesn't tell us the measure of the sides to determine the perimeter - insufficient

From(1) and (2) together

From (2) we know that there is only one of such parallelograms that has an angle 45 to the base.
Base = 10. One angle = 45 degrees
In Parallelogram opposite angles are equal ..
Hence there are 2*45 degrees
Sum of all angles = 360
Thus 2*45 + 2 x = 360 which gives each other angle as 135 degrees


Note: If we are able to draw the figure on given DS question it implies conditions given are sufficient.

Problem Solving - 20

What is the area of the trapezoid pictured to the right?

(A) 22.5

(B) 24

(C) 27

(D) 45

(E) 54


Answer: A

Since the triangle is a right isosceles triangle, the other side of the triangle (the height of the trapezoid) must also be 3. The top base if the trapezoid must be 6 since it is the opposite side of a rectangle.

The area of a trapezoid = 1/2 (sum of parallel sides) × (distance between the parallel sides)
= 1/2 (9 + 6) (3) = 22.5

The area can also be found here by breaking up the figure into a rectangle (area of 6 × 3 = 18) and a triangle (area of ½ × 3 × 3 = 4.5) and adding these two areas.

The correct answer is A.

Manhattan Challenge Problem of the week! - 02/04/07

Triangle A has one side of length x. If (x^8) ^ 1/2 = 81 , what is the perimeter of Triangle A?

1) Triangle A has sides whose lengths are consecutive integers
2) Triangle A is NOT a right triangle

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Answer : C

OE - By simplifying the equation given in the question stem, we can solve for x as follows:

(x^8) ^ 1/2 = 81

x^4 = 81

x = 3

Thus, we know that one side of Triangle A has a length of 3.

Statement (1) tells us that Triangle A has sides whose lengths are consecutive integers. Given that one of the sides of Triangle A has a length of 3, this gives us the following possibilities: (1, 2, 3) OR (2, 3, 4) OR (3, 4, 5).

However, the first possibility is NOT a real triangle, since it does not meet the following condition, which is true for all triangles: The sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. Since 1 + 2 is not greater than 3, it is impossible for a triangle to have side lengths of 1, 2 and 3.

Thus, Statement (1) leaves us with two possibilities. Either Triangle A has side lengths 2, 3, 4 and a perimeter of 9 OR Triangle A has side lengths 3, 4, 5 and a perimeter of 12. Since there are two possible answers, Statement (1) is not sufficient to answer the question.

Statement (2) tells us that Triangle A is NOT a right triangle. On its own, this is clearly not sufficient to answer the question, since there are many non-right triangles that can be constructed with a side of length 3.

Taking both statements together, we can determine the perimeter of Triangle A.

From Statement (1) we know that Triangle A must have side lengths of 2, 3, and 4 OR side lengths of 3, 4, and 5. Statement (2) tells us that Triangle A is not a right triangle; this eliminates the possibility that Triangle A has side lengths of 3, 4, and 5 since any triangle with these side lengths is a right triangle (this is one of the common Pythagorean triples). Thus, the only remaining possibility is that Triangle A has side lengths of 2, 3, and 4, which yields a perimeter of 9.

The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Probelm Solving - 17

On the coordinate plane (6, 2) and (0, 6) are the endpoints of the diagonal of a square. What is the distance between point (0, 0) and the closest vertex of the square?

(A) 1/sqrt (2)
(B) 1
(C) sqrt (2)
(D) sqrt (3)
(E) 2*sqrt (3)

Answer : C

Midpoint of the diagonal = (3,4)
Slope of the diagonal = -2/3
Slope of the other diagonal is the negative reciprocal of the first diagonal = 3/2

Let one of the vertices = (a,b).

The midpoint (diagonal) and the slope (diagonal) gives us the 1st equation = (y-4)/(x-3) = 3/2

The length of the diagonal = 2*sqrt13.

Calculate the length of the diagonal from the midpoint to the vertex (1/2 of diagonal) .

This gives us our 2nd equation = (x-3)^2 + (y-4)^2 = 13

Solving both the equations :

(x,y) = (1,1)

Thus, the distance is sqrt 2

Problem Solving - 7

Eight points lie on the circumfrence of a circle.What is the positive diff b/w th no. of triangles and the no. of quadrilaterals that can be formed by connecting these points?

a. 8

b. 14

c. 56

d. 70

e. 1,344

Answer - B

Since the points are on the circumference of the circle thus no 3 or 4 points will be colinear, thus all points can be used to draw a triangle or quadilateral

Hence 8C4 = 70 and 8C3 = 56 (we make use of combination here because here the order of points do not matter)

Taking the difference we get 70 - 56 = 14.