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JEE Properties of Triangles PYQs with Solutions

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

64 questionsIOQMAnswers + solutions

About Properties of Triangles PYQs

Properties of Triangles has 64 previous year questions in the Quantrex Academy JEE Main & Advanced IOQM bank. The 64 questions in this list come from Olympiad workbook (64). Each question opens on its own page with the options, the correct answer and the worked solution where available.

Properties of Triangles questions with solutions

1

The incircle Γ 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 Γ and the sides A B and A

Olympiad workbook Solution View → 2

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

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

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

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, r_1 , and r_2 be the inradii of triangles A B C, A B D and A C D

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

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

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

Olympiad workbook Solution View → 14

Let ABC be a triangle with sides 51,52,53 . Let Ω denote the incircle of ABC . Draw tangents to Ω which are parallel to the sides of ABC . Let r_1, r_2, r_3 be the inradii of the

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

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

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

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

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^ ∘ ≈

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

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 G_1, G_2 and G_3 be the centroids of the triangles HBC, HCA and HAB respectively. If the area of triangle

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 I_1 and A I_2 meet B C in E and F

Olympiad workbook Solution View → 31

Let ABC be a triangle and let Ω be its circumcircle. The internal bisectors of angles A, B and C intersect Ω at A_1, ~B_1 and C_1 respectively and the internal bisectors of angles

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

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 Δ)/(x)=y+(2 Δ)/(y) , where Δ is the area of the triangle. If x=60, y=63 , what is the length of the largest side of the

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, r_1 , and r_2 be the inradii of triangles A B C, A B D and A C D

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

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 Γ 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 Γ and the sides A B and A

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

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

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

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

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

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

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

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

Olympiad workbook Solution View → 52

Let ABC be a triangle with sides 51,52,53 . Let Ω denote the incircle of ABC . Draw tangents to Ω which are parallel to the sides of ABC . Let r_1, r_2, r_3 be the inradii of the

Olympiad workbook Solution View → 53

Let ABC be a triangle and let Ω be its circumcircle. The internal bisectors of angles A, B and C intersect Ω at A_1, ~B_1 and C_1 respectively and the internal bisectors of angles

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

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

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^ ∘ ≈

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 G_1, G_2 and G_3 be the centroids of the triangles HBC, HCA and HAB respectively. If the area of triangle

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 I_1 and A I_2 meet B C in E and F

Olympiad workbook Solution View →
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Related JEE IOQM chapters

Permutation and Combination (112)Basics of Mathematics (88)Sequences and Series (38)Quadratic Equation (38)Circle (30)Straight Lines (24)Functions (20)Binary Numbers (8)Vector Algebra (6)Binomial Theorem (6)

Properties of Triangles in other exams

BITSAT Properties of Triangles (7)