# Triangles | Problems

## Problems in Triangles

1. State and prove Basic Proportionality theorem.
Or,
Prove that if a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio.
2. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
3. Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
4. State and prove converse of Pythagoras theorem.
Or,
Prove that in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the frst side is a right angle.
5. If one diagonal of a trapezium divides the other diagonal in the ratio 1 : 2, prove that one of the parallel sides is double the other.
6. In the fgure DE || BC. Find x.
7. If D, E are points on the sides AB and AC of a ∆ABC such that AD = 6 cm, BD = 9 cm, AE = 8 cm EC = 12 cm. Prove that DE || BC.
8. In the fgure, DE || OQ and DF || OR. Show that EF || QR.
9. In the fgure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
10. In the fgure, two triangles ABC and DBC are on the same base BC in which ∠A = ∠D = 90°. If CA and BD meet each other at E, show that AE × CE = BE × DE.
11. In the fgure, PQR and SQR are two triangles on the same base QR. If PS intersect QR at O, then show that: $\inline \dpi{120} \frac { \operatorname { ar } ( P Q R ) } { \operatorname { ar } ( S Q R ) } = \frac { P O } { S O }$.
12. In the given fgure, E is a point on side CB produced of an isosceles ∆ABC with AB = BC. If AD ⊥ BC and EF ⊥ AC, prove that ∆ABD ~ ∆ECF.
13. In the fgure, DE || AC and DF || AE. Prove that $\inline \dpi{120} \frac{EF}{BF}=\frac{EC}{BE}$
14. In the fgure, XN || CA and XM || BA. T is a point on CB produced. Prove that TX2 = TB×TC.
15. Two poles of height 10 m and 15 m stand vertically on a plane ground. If the distance between their feet is 5√3 m, fnd the distance between their tops.
16. The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus.
17. In the fgure, D, E, F, are mid-points of sides BC, CA, AB respectively of ∆ABC. Find the ratio of area of ∆DEF to area of ∆ABC.
18. In the fgure, PA, QB and RC are prependiculars to AC. Prove that $\inline \dpi{120} \frac{1}{x}+\frac{1}{y}=\frac{1}{z}$.
19. Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that EL = 2BL.
20. ABCD is a quadrilateral. P, Q, R, S are the points of trisection of the sides AB, BC, CD and DA respectively. Prove that PQRS is a parallelogram.
21. In an equilateral triangle ABC, D is a point on side BC such that 3BD = BC. Prove that 9 AD2 = 7 AB2.
22. PQR is a triangle in which QM ⊥ PR and PR2 – PQ2 = QR2, prove that QM2 = PM × MR.
23. Triangle ABC is right angled at B and D is mid-point of BC. Prove that : AC2 = 4AD2 – 3AB2.
24. In fgure, ABC is right triangle right angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB. Prove that : (i) cp = ab (ii) $\inline \dpi{120} \frac { 1 } { p ^ { 2 } } = \frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } }$
25. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ∆PQR. Show that ∆ABC ~ ∆PQR.
26. In the fgure, DEFG is a square and ∠BAC = 90°. Show that DE2 = BD × EC.
27. O is a point in the interior of rectangle ABCD. If O is joined to each of the vertices of the rectangle, prove that OB2 + OD2 = OA2 + OC2.
28. In ∆ABC, AD is a median and E is mid-point of AD. If BE is produced, it meets AC at F. Show that $\inline \dpi{120} A F = \frac { 1 } { 3 } A C$
29. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
30. Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.
31. Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights.
32. If D is a point on the side AB of ∆ABC such that AD : DB = 3 : 2 and E is a point on BC such that DE || AC. Find the ratio of areas of ∆ABC and ∆BDE.
33. In the fgure, ∆PQR is right angled at Q and the points S and T trisect the side QR. Prove that 8PT2 = 3PR2 + 5PS2
34. An aeroplane leaves an airport and ﬂies due north at a speed of 500 km per hour. At the same time, another aeroplane leaves the same airport and ﬂies due west at a speed of 1200 km per hour. How far apart will be the two planes after 2 hours?