GEOMETRY FOR PRMO

Special section

triangles

circles

1)In a rectangle ABCD, E is the midpoint of AB : F is point on AC such that BF is perpendicular to AC and FE perpendicular to BD. Suppose BC =  8√3. Find AB.(PRMO 2017)

Ans. 24

2)If in a quadrilateral ABCD, AC bisects angles A and C, show that AC is perpendicular to BD

 In quadrilateral ABCD , 

    AC and BD are diagonals . So let they intersect at O.

     In triangles ABC and ADC,

         angle CAB = angle CAD

         angle  ACB = angle ACD

               AC = AC

               Therefore, triangles ABC and ADC are congruent.

               NOW,

                       AB = AD (corresponding parts of congruent triangles)

                      In triangle ABD,

                       AB = AD

      So, triangle ABD is isosceles.  As we know the angle bisector of the vertical angle of an isosceles triangle 

                        is also the perpendicular bisector of the base.

 So AC intersects BD at right angle

3)If two straight lines AB and CD meet at O , and there is another line XY . prove in general that there are two points on XY that are equidistant from the two lines AB  and CD . WHEN  is there only one such line .

We know that any point lying on angle bisector of two lines is equidistant from both the lines.

Now we know that the third line would definitely intersect both the angle bisectors until it is parallel to one of the bisector.

If it is parallel to one of the bisectors, then there would be only 1 which would be equidistant from both the lines

4)Let P be an interior point of a triangle ABC whose side lengths are 26, 65, 78. The line through P parallel

to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line

through P parallel to AB meets CA in S and CB in T. If KL, MN, ST are of equal lengths, find this common

length. PRMO 2017

5)Let ABCD be a rectangle and let E and F be points on CD and BC respectively such that area (ADE) =16, area (CEF) = 9 and area (ABF) = 25. What is the area of triangle AEF ?PRMO 2017

6)Let AB and CD be two parallel chords in a circle with radius 5 such that the centre O lies between these

chords. Suppose AB = 6, CD = 8. Suppose further that the area of the part of the circle lying between the

chords  AB and CD is

where m, n, k are positive integers with gcd(m, n, k) = 1. What is the

value of m + n + k?PRMO 2017

7)PRMO 2017

ANSWER 

8)Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.PRMO 2017

test yourself

triangles

Part 2

In a quadrilateral ABCD, which is not a trapezium, it is known that ∠DAB = ∠ABC = ° 60 . Moreover,

∠CAB = ∠CBD . Then.

(A) AB = BC + CD (B) AB = AD + CD (C) AB = BC +AD (D) AB = AC + AD

Ans D