Online CAT Coaching: A few interesting True/False questions from Geometry

State whether the following statements are true or false

1. A parallelogram that circumscribes a circle has to be a square
2. A trapezium inscribed in a circle has to be an isosceles trapezium
3. Orthocenter of a triangle can lie outside the triangle
4. Triangle with sides a, b and c has the relationship a^2 + b^2 > c^2, the triangle has to be acute-angled.
5. Diagonals of a parallelogram are angle bisectors of the angles of a parallelogram.

Scroll down for answers and explanation
























1. A parallelogram that circumscribes a circle has to be a square: FALSE

In a parallelogram, opposite sides are equal. In a quadrilateral, the sums of pairs of opposite sides are equal. So, a parallelogram that circumscribes a circle should have all 4 of its sides equal. Or, it should be a Rhombus; it need not be a square.

2. A trapezium inscribed in a circle has to be an isosceles trapezium: TRUE

An isosceles trapezium is a symmetric diagram. The two base angles should be equal and the two top angles should be equal. So, a trapezeium where the base angles were equal would be an isosceles trapezium.

In any cyclic quadrilateral, opposite angles would be supplementary. In a trapezium, co-interior angles between the parallel lines would be supplementary. So, if we took a trapezium ABCD with AB parallel to CD inscribed in a circle. Angle A and Angle D would be supplementary (co-interior angles). And Angle A and Angle C would be supplementary (opposite angles of a cyclic quadrilateral). Or angle B would be equal to angle C. Ergo, isosceles trapezium.

3. Orthocenter of a triangle can lie outside the triangle: TRUE

For any obtuse-angled triangle, two of the altitudes would lie outside the triangle, and would intersect at a point outside the triangle. So, the orthocenter can lie outside the triangle.

4. Triangle with sides a, b and c has the relationship a^2 + b^2 > c^2, the triangle has to be acute-angled: FALSE

Let us take triangle with sides 2, 3 and 4. 4^2 + 3^2 > 2^2. But  as 2^2 + 3^3 < 4^2, the triangle is obtuse-angled. Is a^2 + b^2 > c^2, we can say angle C is acute-angled. We cannot say all three angles are acute-angled. One can use cosine rule also for having a go at this question (though it should be considered inelegant)


5. Diagonals of a parallelogram are angle bisectors of the angles of a parallelogram: FALSE

Diagonals of a parallelogram bisect each other. They need not bisect the angles of the parallelogram. Imagine this, if we took a rectangle and studied its diagonals. if the diagonals bisected each other, the angle between diagonal and a side would be 45 degrees. Or, we would end up having a square. So, any rectangle that was not a square would have diagonals that were not angle bisectors. So, diagonals of a parallelogram NEED NOT be angle bisectors of the angles of a parallelogram.

CAT Geometry

Question

 x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4, find z.

A.    2

B.    3

C.    1

D.    More than one possible value of z exists

Answer:  Choice B

Explanatory Answer:

xy = 4

xy could be 2 x 2 or 4 x 1

221

222            These are the possible triangles 

223

441

22x will be a triangle if x is 1, 2 or 3 (trial and error)

44x is a triangle only if x is 1.

Ø  221 is acute.  1² + 2²> 2²

Ø  222 is equilateral. So acute.

Ø  223 is obtuse.  2² + 2²< 3²

Ø  144 is acute.  1² + 4²> 4² 

Only triangle 223 is obtuse.  Hence, the third side has to be 3.

Answer Choice (B)

CAT Geometry

Question

Sides of a triangle are 6, 10 and x for what value of x is the area of the D the maximum?

A.    8 cms

B.    9 cms

C.    12 cms

D.    None of these

Correct Answer: D

Explanatory Answer

Side of a triangle are. 6, 10, x.

Area = 1/2 x 6 x 10 sin ÐBAC.

Area is maximum, when ÐBAC = 90^o

x = sqrt(100 + 36) = sqrt(136)

There is a more algebraic method using hero’s formula. Try that also.

Answer Choice (D)

Solutions to CAT Geometry Data Sufficiency Questions

Have given below the solutions to the questions on geometry DS. The solutions are courtesy Vimal Gopinath (person incharge of 2IIM Bengaluru)

Qn 1: Is triangle ABC obtuse angled?
I) a^2 + b^2 > c^2 - Not enough. We don’t have info about b^2 or a^2
II) The circumcenter of the triangle does not lie inside the triangle - Not enough. The triangle could be right-angled as well.
Combination is also not enough, it is valid for both right angled and obtuse angled triangles.
D

Qn 2: Do the two circles with centers A and B and radii R and r intersect each other
I) AB > R - r – Not enough. May intersect, may be “parallel” or "disjoint"
II) AB > R + r – Sufficient. Circles cant intersect. Have to be separate.
A

Qn 3: Trapezium ABCD is such that AB is parallel to CD. Is this trapezium anisosceles trapezium?
I) Angle B and D are supplementary Sufficient. If two base angles are equal, then the trapezium has to be isosceles.
II) The quadrilateral is inscribed inside a circle. Sufficient. All trapeziums inscribed in circles have to be isosceles. (Think about the proof for this)
C

Qn 4: Circle C has center O, and a chord AB such that angle AOB = 80 degrees.Does point E lie inside the circle

I) Angle AEB > 50 degrees Insufficient.E could lie on the minor segment ADB or slightly outside or slightly inside the circle.
II) Angle AEB < 30 degrees. Sufficient. All the angles inside the circle will be in the range from 40 – 140 degrees. Anything less than 40 will have to be outside the circle.)
A

CAT Geometry Data Sufficiency questions

For the following questions,

Mark A) If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone
Mark B) If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together
Mark C) If the question can be answered with either statement alone
Mark D) If the question cannot be answered with the information provided

Qn 1: Is triangle ABC obtuse angled
I) a^2 + b^2 > c^2
II) The circumcenter of the triangle does not lie inside the triangle

Qn 2: Do the two circles with centers A and B and radii R and r intersect each other
I) AB > R - r
II) AB > R + r

Qn 3: Trapezium ABCD is such that AB is parallel to CD. Is this trapezium an isosceles trapezium?
I) Angle B and D are supplementary
II) The quadrilateral is inscribed inside a circle

Qn 4: Circle C has center O, and a chord AB such that angle AOB = 80 degrees. Does point E lie inside the circle
I) Angle AEB > 50 degrees
II) Angle AEB < 30 degrees

CAT Geometry Solutions

Have given below the solutions to the questions on basic geometry

1. Perimeter of a triangle with integer sides is equal to 15. How many such triangles are possible?

This is just a counting question, with the caveat that sum of two sides should be greater than the third. Let us assume a < b < c

a = 1, Possible triangle 1, 7, 7
a = 2, possible triangle 2, 6, 7
a = 3, possible triangles 3, 6, 6 and 3, 5, 7
a = 4, possible triangles 4, 4, 7 and 4, 5, 6

Again, from comments,
a = 5, possible triangle is 5,5,5,

There are totally 7 triangles possible

2. Triangle ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?

xz = 12

x,z can be 1, 12 or 2, 6 or 3, 4

Possible triangles
1-12-12
2-6-5; 2-6-6; 2-6-7
3-4-2; 3-4-3; 3-4-5; 3-4-6.

As pointed out in the comments section, I have missed the triangle 3-4-4.

There are totally 9 triangles.

3. Triangle has sides a^2, b^2 and c^2. Then the triangle with sides a, b, c has to be - a) Right angled b) Acute-angled c) Obtuse angled d) can be any of these three

Assuming a < b < c, we have a^2 + b^2 > c^2. This implies the triangle with sides a, b, c has to be acute-angled.

P.S: Big thanks to 'maniac' for pointing out the errors