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Properties of Triangles — JEE Main & Advanced IOQM PYQs

64 previous year questions from Properties of Triangles with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

64 questionsIOQMSolutions on every page
1

The incircle of a scalene triangle A B C touches B C at D, C A at E and A B at F . Let r_A be the radius of the circle inside A B C which is tangent to and the sides A B and A C .

Olympiad workbook Solution View → 2

The sides x and y of a scalene triangle satisfy x+ 2 x =y+ 2 y , where is the area of the triangle. If x=60, y=63 , what is the length of the largest side of the triangle?

Olympiad workbook Solution View → 3

In a triangle A B C , the median A D divides B A C in the ratio 1: 2 . Extend A D to E such that E B is perpendicular A B . Given that B E=3, B A=4 , find the integer nearest to B

Olympiad workbook Solution View → 4

Let A B C be a triangle let D be a point on the segment B C such that A D=B C . Suppose C A D=x^ , A B C=y^ and A C B=z^ and x, y, z are in an arithmetic progression in that order

Olympiad workbook Solution View → 5

Two sides of a regular polygon with n sides, when extended, meet at an angle of 28^ . What is the smallest possible value of n ?

Olympiad workbook Solution View → 6

Let A B C be an isosceles triangle with A B=A C and incentre I . If A I=3 and the distance from I to B C is 2 , what is the square of the length of B C ?

Olympiad workbook Solution View → 7

Let A B C be an acute-angled triangle and P be a point in its interior. Let P_A, P_B , and P_C be the images of P under reflection in the sides B C, C A and A B , respectively. If

Olympiad workbook Solution View → 8

Let A B C be a triangle with B A C=90^ and D be the point on the side B C such that A D B C . Let r, r₁ , and r₂ be the inradii of triangles A B C, A B D and A C D , respectively.

Olympiad workbook Solution View → 9

The sides of a triangle are x, 2 x+1 and x+2 for some positive rational number x . If one angle of the triangle is 60^ , what is the perimeter of the triangle?

Olympiad workbook Solution View → 10

Let A B C be a triangle with A B=5, A C=4, B C=6 . The internal angle bisector of C intersects the side A B at D . Points M and N are taken on sides B C and A C , respectively, suc

Olympiad workbook Solution View → 11

In a triangle A B C, B A C=90^ . Let D be the point on B C such that A B+B D=A C+C D . Suppose B D: D C=2: 1 . If A C A B = m+ p n , where m, n are relatively prime positive intege

Olympiad workbook Solution View → 12

In a trapezoid A B C D , the internal bisector of angle A intersects the base B C (or its extension) at the point E . Inscribed in the triangle A B E is a circle touching the side

Olympiad workbook Solution View → 13

The ex-radii of a triangle are 10 1 2 , 12 and 14 . If the sides of the triangle are the roots of the cubic x^3-p x^2+q x-r=0 , where p, q, r are integers, find the integer nearest

Olympiad workbook Solution View → 14

Let ABC be a triangle with sides 51,52,53 . Let denote the incircle of ABC . Draw tangents to which are parallel to the sides of ABC . Let r ₁, r ₂, r ₃ be the inradii of the three

Olympiad workbook Solution View → 15

In a quadrilateral A B C D , it is given that A B=A D=13, B C=C D=20, B D=24 . If r is the radius of the circle inscribable in the quadrilateral, then what is the integer closest t

Olympiad workbook Solution View → 16

Let A B C be a triangle with A B=A C . Let D be a point on the segment B C such that B D=48 1 61 and D C=61 . Let E be a point on A D such that C E is perpendicular to A D and D E=

Olympiad workbook Solution View → 17

In a triangle A B C , let E be the midpoint of A C and F be the midpoint of A B . The medians B E and C F intersect at G . Let Y and Z be the midpoints of B E and C F respectively.

Olympiad workbook Solution View → 18

If 15 and 9 are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?

Olympiad workbook Solution View → 19

The centre of the circle passing through the midpoints of the sides of an isosceles triangle A B C lies on the circumcircle of triangle A B C . If the larger angle of triangle A B

Olympiad workbook Solution View → 20

Let A B C be a right-angled triangle with B=90^ . Let the length of the altitude B D be equal to 12 . What is the minimum possible length of A C , given that A C and the perimeter

Olympiad workbook Solution View → 21

In a triangle ABC , it is known that A =100^ and AB = AC . The internal angle bisector BD has length 20 units. Find the length of BC to the nearest integer, given that 10^ 0.174 .

Olympiad workbook Solution View → 22

Let ABC be an isosceles triangle with AB = BC . A trisector of B meets AC at D . If AB , AC and B D are integers and A B-B D=3 , find A C .

Olympiad workbook Solution View → 23

Let x, y, z be positive real number such that x^2+y^2=49, y^2+y z+z^2=36 and x^2+ 3 x z+z^2=25 . If the value of 2 x y+ 3 y z+z x can be written as p q where p, q are integers and

Olympiad workbook Solution View → 24

Let A B C be an equilateral triangle with side length 10. A square P Q R S is inscribed in it, with P on A B , Q, R on B C and S on A C . If the area of the square P Q R S is m+n k

Olympiad workbook Solution View → 25

a, b, c are positive real numbers such that a^2+b^2 =c^2 and a b=c . Determine the value of | (a+b+c)(a+b-c)(b+c-a)(c+a-b) c^2 | .

Olympiad workbook Solution View → 26

Triangles ABC and DEF are such that A = D , AB = DE =17, BC = EF =10 and AC - DF =12 . What is AC + DF ?

Olympiad workbook Solution View → 27

In a triangle ABC , right-angled at A , the altitude through A and the internal bisector of A have lengths 3 and 4, respectively. Find the length of the median through A .

Olympiad workbook Solution View → 28

In a triangle A B C , the median from B to C A is perpendicular to the median from C to A B . If the median from A to B C is 30 , determine (B C^2+C A^2+A B^2 ) / 100 .

Olympiad workbook Solution View → 29

Let A B C be an acute-angled triangle and let H be its orthocentre. Let G₁, G₂ and G₃ be the centroids of the triangles HBC , HCA and HAB respectively. If the area of triangle G ₁

Olympiad workbook Solution View → 30

Let D be an interior point of the side BC of a triangle ABC . Let I ₁ and I ₂ be the incentres of triangles A B D and A C D respectively. Let A I₁ and A I₂ meet B C in E and F resp

Olympiad workbook Solution View → 31

Let ABC be a triangle and let be its circumcircle. The internal bisectors of angles A, B and C intersect at A ₁, ~B ₁ and C ₁ respectively and the internal bisectors of angles A ₁,

Olympiad workbook Solution View → 32

In a triangle ABC , the median AD (with D on BC ) and the angle bisector BE (with E on AC ) are perpendicular to each other. If AD =7 and BE =9 , find the integer nearest to the ar

Olympiad workbook Solution View → 33

The sides of a triangle are x, 2 x+1 and x+2 for some positive rational number x . If one angle of the triangle is 60^ , what is the perimeter of the triangle?

Olympiad workbook Solution View → 34

a, b, c are positive real numbers such that a^2+b^2 =c^2 and a b=c . Determine the value of | (a+b+c)(a+b-c)(b+c-a)(c+a-b) c^2 | .

Olympiad workbook Solution View → 35

Let A B C be an isosceles triangle with A B=A C and incentre I . If A I=3 and the distance from I to B C is 2 , what is the square of the length of B C ?

Olympiad workbook Solution View → 36

The sides x and y of a scalene triangle satisfy x+ 2 x =y+ 2 y , where is the area of the triangle. If x=60, y=63 , what is the length of the largest side of the triangle?

Olympiad workbook Solution View → 37

Let A B C be a triangle with B A C=90^ and D be the point on the side B C such that A D B C . Let r, r₁ , and r₂ be the inradii of triangles A B C, A B D and A C D , respectively.

Olympiad workbook Solution View → 38

Two sides of a regular polygon with n sides, when extended, meet at an angle of 28^ . What is the smallest possible value of n ?

Olympiad workbook Solution View → 39

Let A B C be a triangle with A B=5, A C=4, B C=6 . The internal angle bisector of C intersects the side A B at D . Points M and N are taken on sides B C and A C , respectively, suc

Olympiad workbook Solution View → 40

Let A B C be an acute-angled triangle and P be a point in its interior. Let P_A, P_B , and P_C be the images of P under reflection in the sides B C, C A and A B , respectively. If

Olympiad workbook Solution View → 41

The incircle of a scalene triangle A B C touches B C at D, C A at E and A B at F . Let r_A be the radius of the circle inside A B C which is tangent to and the sides A B and A C .

Olympiad workbook Solution View → 42

In a triangle A B C , the median A D divides B A C in the ratio 1: 2 . Extend A D to E such that E B is perpendicular A B . Given that B E=3, B A=4 , find the integer nearest to B

Olympiad workbook Solution View → 43

Let x, y, z be positive real number such that x^2+y^2=49, y^2+y z+z^2=36 and x^2+ 3 x z+z^2=25 . If the value of 2 x y+ 3 y z+z x can be written as p q where p, q are integers and

Olympiad workbook Solution View → 44

Let A B C be a triangle let D be a point on the segment B C such that A D=B C . Suppose C A D=x^ , A B C=y^ and A C B=z^ and x, y, z are in an arithmetic progression in that order

Olympiad workbook Solution View → 45

Let A B C be an equilateral triangle with side length 10. A square P Q R S is inscribed in it, with P on A B , Q, R on B C and S on A C . If the area of the square P Q R S is m+n k

Olympiad workbook Solution View → 46

The ex-radii of a triangle are 10 1 2 , 12 and 14 . If the sides of the triangle are the roots of the cubic x^3-p x^2+q x-r=0 , where p, q, r are integers, find the integer nearest

Olympiad workbook Solution View → 47

In a trapezoid A B C D , the internal bisector of angle A intersects the base B C (or its extension) at the point E . Inscribed in the triangle A B E is a circle touching the side

Olympiad workbook Solution View → 48

In a triangle A B C , let E be the midpoint of A C and F be the midpoint of A B . The medians B E and C F intersect at G . Let Y and Z be the midpoints of B E and C F respectively.

Olympiad workbook Solution View → 49

In a triangle A B C, B A C=90^ . Let D be the point on B C such that A B+B D=A C+C D . Suppose B D: D C=2: 1 . If A C A B = m+ p n , where m, n are relatively prime positive intege

Olympiad workbook Solution View → 50

Let A B C be a right-angled triangle with B=90^ . Let the length of the altitude B D be equal to 12 . What is the minimum possible length of A C , given that A C and the perimeter

Olympiad workbook Solution View → 51

Let A B C be a triangle with A B=A C . Let D be a point on the segment B C such that B D=48 1 61 and D C=61 . Let E be a point on A D such that C E is perpendicular to A D and D E=

Olympiad workbook Solution View → 52

Let ABC be a triangle with sides 51,52,53 . Let denote the incircle of ABC . Draw tangents to which are parallel to the sides of ABC . Let r ₁, r ₂, r ₃ be the inradii of the three

Olympiad workbook Solution View → 53

Let ABC be a triangle and let be its circumcircle. The internal bisectors of angles A, B and C intersect at A ₁, ~B ₁ and C ₁ respectively and the internal bisectors of angles A ₁,

Olympiad workbook Solution View → 54

In a triangle ABC , the median AD (with D on BC ) and the angle bisector BE (with E on AC ) are perpendicular to each other. If AD =7 and BE =9 , find the integer nearest to the ar

Olympiad workbook Solution View → 55

In a quadrilateral A B C D , it is given that A B=A D=13, B C=C D=20, B D=24 . If r is the radius of the circle inscribable in the quadrilateral, then what is the integer closest t

Olympiad workbook Solution View → 56

If 15 and 9 are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?

Olympiad workbook Solution View → 57

In a triangle ABC , right-angled at A , the altitude through A and the internal bisector of A have lengths 3 and 4, respectively. Find the length of the median through A .

Olympiad workbook Solution View → 58

In a triangle A B C , the median from B to C A is perpendicular to the median from C to A B . If the median from A to B C is 30 , determine (B C^2+C A^2+A B^2 ) / 100 .

Olympiad workbook Solution View → 59

The centre of the circle passing through the midpoints of the sides of an isosceles triangle A B C lies on the circumcircle of triangle A B C . If the larger angle of triangle A B

Olympiad workbook Solution View → 60

Let ABC be an isosceles triangle with AB = BC . A trisector of B meets AC at D . If AB , AC and B D are integers and A B-B D=3 , find A C .

Olympiad workbook Solution View → 61

In a triangle ABC , it is known that A =100^ and AB = AC . The internal angle bisector BD has length 20 units. Find the length of BC to the nearest integer, given that 10^ 0.174 .

Olympiad workbook Solution View → 62

Triangles ABC and DEF are such that A = D , AB = DE =17, BC = EF =10 and AC - DF =12 . What is AC + DF ?

Olympiad workbook Solution View → 63

Let A B C be an acute-angled triangle and let H be its orthocentre. Let G₁, G₂ and G₃ be the centroids of the triangles HBC , HCA and HAB respectively. If the area of triangle G ₁

Olympiad workbook Solution View → 64

Let D be an interior point of the side BC of a triangle ABC . Let I ₁ and I ₂ be the incentres of triangles A B D and A C D respectively. Let A I₁ and A I₂ meet B C in E and F resp

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