In a triangle ABC, AB = 16, AC = 9. Find the length of the altitude to BC if the circumradius is 9. Answer: 8 Explanation: Consider the figure below. We know that, Area of a triangle = abc/4R where a,b,c are the sides of the triangle and R is the circumradius. Hence, (AB x BC x AC)/4R = (1/2) x BC x AD (16 x BC x 9)/36 = (1/2) x BC x AD Therefore, AD = 8. … [Read more...] about Geometry Triangle – Equating Areas

# GRE Geometry, Solid Geometry, Coordinate Geometry

## Geometry – Obtuse Angled Triangles

How many obtuse-angled triangles can you form with 10,24,x as the sides where x is an integer? Answer: 14 Explanation: For three numbers to form an obtuse-angled triangle the square of the largest number should be greater than the sum of the squares of the other two numbers. For three numbers to form a triangle the sum of any two numbers should be greater than the … [Read more...] about Geometry – Obtuse Angled Triangles

## Geometry Circles – Angle in Same Segment

Q,R,S,T are four points on the circumference of a circle. QT is the perpendicular bisector of RS. PQ is a tangent drawn to the circle from an external point P. If angle(SQP) = 20, find angle(TRS). Answer: 70 degrees Explanation: Consider the diagram below. QT is the perpendicular bisector of a chord RS. Hence QT has to pass through the centre of the … [Read more...] about Geometry Circles – Angle in Same Segment

## Cyclic Quadrilateral Circles

In a cyclic quadrilateral ABCD, the diagonals intersect at E. If AE = 4 and EC = 9, what is the minimum length of diagonal BD? Answer: 12 Explanation We know that when two chords of a circle intersect at a point, AE*EC = BE*ED Hence, BE*ED = 4*9 = 36. Given the product of two quantities their sum is minimum when the two quantities are equal. Hence … [Read more...] about Cyclic Quadrilateral Circles

## Quantitative Comparison – Coordinate Geometry

Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:(A) Quantity A is greater.(B) Quantity B is greater.(C) The two quantities are equal.(D) The relationship cannot be determined from the information given.A circle with radius 5 and passing through the … [Read more...] about Quantitative Comparison – Coordinate Geometry