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Geometry Fundas for CAT Quantitative Aptitude Preparation – Part 1

Sunday, July 26th, 2020

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I got a lot of feedback via emails and texts that people are looking for a post on geometry. I have been avoiding it for sometime because of two main reasons:

a)      It is not one of my strong areas.

b)      It takes a lot of time to draw the diagrams that are sometimes required to explain the fundas.

The questions on geometry are the trickiest and consumes the maximum amount of time as compared to the questions on other topics in Quantitative aptitude that is why, I have compiled a list of fundas that you might find helpful in solving CAT level questions. I am splitting those in two posts so that one post does not become too long / intimidating. In this post, we will discuss Geometry fundas related to lines, triangles, parallelograms, trapeziums, polygons, etc.

You might find some of them very simple or ideas that are obvious to you. If that is the case, be glad that your preparation is up to the mark. If not, then be glad you got them in time. (Yes – I am inspired by two-face J)

Funda 1:

The ratio of intercepts formed by a transversal intersecting three parallel lines is equal to the ratio of corresponding intercepts formed by any other transversal.

Funda 2:

Centroid and Incenter will always lie inside the triangle. About the other points:

–          For an acute angled triangle, the Circumcenter and the Orthocenter will lie inside the triangle.

–          In case of an obtuse angled triangle, the Circumcenter and the Orthocenter will lie outside the triangle.

–          and right-angled triangle, the Circumcenter will lie at the midpoint of the hypotenuse and the Orthocenter will lie at the vertex at which the angle is 90°.

 Funda 3:

The orthocenter, centroid, and circumcenter always lie on the same line known as Euler Line.

–          The orthocenter is twice as far from the centroid as the circumcenter is.

–          If the triangle is Isosceles then the incenter lies on the same line.

–          If the triangle is equilateral, all four are the same point.

Funda 4:

Appolonius’ Theorem {AD is the median}

AB²Â + AC²Â = 2 (AD²Â + BD²)

 Funda 5: For cyclic quadrilaterals –

Area =  where s is the semi perimeter

Also, Sum of product of opposite sides = Product of diagonals

•  ac + bd = PR x QS

Funda 6:

If a circle can be inscribed in a quadrilateral, its area is given by =

Funda 7:

Parallelograms

• A parallelogram inscribed in a circle is always a Rectangle. A parallelogram circumscribed about a circle is always a Rhombus. So, a parallelogram that can be circumscribed about a circle and in which a circle can be inscribed will be a Square.
  • Each diagonal divides a parallelogram in two triangles of equal area.
  • Sum of squares of diagonals = Sum of squares of four sides

AC²Â + BD²Â = AB²Â + BC²Â + CD²Â + DA²

• A Rectangle is formed by intersection of the four angle bisectors of a parallelogram.
• From all quadrilaterals with a given area, the square has the least perimeter. For all quadrilaterals with a given perimeter, the square has the greatest area.

 

Funda 8:Trapeziums

• Sum of the squares of the length of the diagonals = Sum of squares of lateral sides + 2 Product of bases.

AC²Â + BD²Â = AD²Â + BC²Â + 2 x AB x CD

• If a trapezium is inscribed in a circle, it has to be an isosceles trapezium.
• If a circle can be inscribed in a trapezium, Sum of parallel sides = Sum of lateral sides.

 

Funda 9:

• A regular hexagon is considered as a combination of six equilateral triangles.
• All regular polygons is considered as a combination of ‘n’ isosceles triangles.

I will wrap up this post here. In my next and final post on Geometry we will discuss fundas related to circles (specifically – common tangents), solid figures, mensuration and co-ordinate geometry.

Other posts related to Quantitative Aptitude – Geometry

Geometry Basics for CAT – Triangle related questions and problems
Mensuration Basics and 3-Dimensional Geometry Concepts for CAT

CAT Questions related to Quantitative Aptitude – Geometry

All questions from CAT Exam Quantitative Aptitude – Geometry
Quantitative Aptitude – Geometry – Triangles – Q1: Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB.
Quantitative Aptitude – Geometry – Triangles – Q2: Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively.
Quantitative Aptitude – Geometry – Triangles – Q3: From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC.
Quantitative Aptitude – Geometry – Circles – Q1: ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees
Quantitative Aptitude – Geometry – Circles – Q2: Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC.
Quantitative Aptitude – Geometry – Coordinate – Q1: The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Quantitative Aptitude – Geometry – Coordinate – Q2: The shortest distance of the point (½, 1) from the curve y = |x -1| + |x + 1| is
Quantitative Aptitude – Geometry – Mensuration – Q1: The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm
Quantitative Aptitude – Geometry – Mensuration – Q2: A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically.
Quantitative Aptitude – Geometry – Mensuration – Q3: A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27.
Quantitative Aptitude – Geometry – Polygon – Q1: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is
 

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