# Triangles | Problems

## Problems in Triangles

- 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. - Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
- Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- 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. - 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.
- In the fgure DE || BC. Find x.

- 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.
- In the fgure, DE || OQ and DF || OR. Show that EF || QR.

- 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.

- 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.

- In the fgure, PQR and SQR are two triangles on the same base QR. If PS intersect QR at O, then show that: .

- 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.

- In the fgure, DE || AC and DF || AE. Prove that

- In the fgure, XN || CA and XM || BA. T is a point on CB produced. Prove that TX
^{2}= TB×TC.

- 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.
- The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus.
- 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.

- In the fgure, PA, QB and RC are prependiculars to AC. Prove that .

- 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.
- 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.
- In an equilateral triangle ABC, D is a point on side BC such that 3BD = BC. Prove that 9 AD
^{2}= 7 AB^{2}. - PQR is a triangle in which QM ⊥ PR and PR
^{2}– PQ^{2}= QR^{2}, prove that QM^{2}= PM × MR. - Triangle ABC is right angled at B and D is mid-point of BC. Prove that : AC
^{2}= 4AD^{2}– 3AB^{2}. - 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)

- 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.
- In the fgure, DEFG is a square and ∠BAC = 90°. Show that DE
^{2}= BD × EC.

- O is a point in the interior of rectangle ABCD. If O is joined to each of the vertices of the rectangle, prove that OB
^{2}+ OD^{2}= OA^{2}+ OC^{2}. - In ∆ABC, AD is a median and E is mid-point of AD. If BE is produced, it meets AC at F. Show that
- Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
- 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.
- Two isosceles triangles have equal vertical angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights.
- 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.
- In the fgure, ∆PQR is right angled at Q and the points S and T trisect the side QR. Prove that 8PT
^{2}= 3PR^{2}+ 5PS^{2}

- 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?