## Important Questions for Class 10 Maths Chapter 6 Triangles

**Question 1. **If ∆ABC ~ ∆PQR, perimeter of ∆ABC = 32 cm, perimeter of ∆PQR = 48 cm and PR = 6 cm, then find the length of AC. (2012)

**Question 2. **∆ABC ~ ∆DEF. If AB = 4 cm, BC = 3.5 cm, CA = 2.5 cm and DF = 7.5 cm, find the perimeter of ∆DEF. (2012, 2017D)

**see also: Class 10 Maths (Question Bank)**

**Question 3.** If ∆ABC ~ ∆RPQ, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm and PQ = 10, then find QR. (2014)

**Question 4. **In ∆DEW, AB || EW. If AD = 4 cm, DE = 12 cm and DW = 24 cm, then find the value of DB. (2015)

**Question 5. **In ∆ABC, DE || BC, find the value of x. (2015)

Question 6.

In the given figure, if DE || BC, AE = 8 cm, EC = 2 cm and BC = 6 cm, then find DE. (2014)

Question 7.

In the given figure, XY || QR, PQXQ=73 and PR = 6.3 cm, find YR. (2017OD)

Question 8.

The lengths of the diagonals of a rhombus are 24 cm and 32 cm. Calculate the length of the altitude of the rhombus. (2013)

Question 9.

If PQR is an equilateral triangle and PX ⊥ QR, find the value of PX^{2}. (2013)

Question 10.

The sides AB and AC and the perimeter P, of ∆ABC are respectively three times the corresponding sides DE and DF and the perimeter P, of ∆DEF. Are the two triangles similar? If yes, find ar(△ABC)ar(△DEF) (2012)

Question 11.

In the figure, EF || AC, BC = 10 cm, AB = 13 cm and EC = 2 cm, find AF. (2014)

Question 12.

X and Y are points on the sides AB and AC respectively of a triangle ABC such that AXAB=14, AY = 2 cm and YC = 6 cm. Find whether XY || BC or not. (2015)

Question 13.

In the given figure, ∠A = 90°, AD ⊥ BC. If BD = 2 cm and CD = 8 cm, find AD. (2012; 2017D)

Question 15.

A 6.5 m long ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall. Find the height of the wall where the top of the ladder touches it. (2015)

Question 16.

In the figure ABC and DBC are two right triangles. Prove that AP × PC = BP × PD. (2013)

Question 17.

In the given figure, QA ⊥ AB and PB ⊥ AB. If AO = 20 cm, BO = 12 cm, PB = 18 cm, find AQuestion (2017OD)

Question 18.

In the given figure, CD || LA and DE || AC. Find the length of CL if BE = 4 cm and EC = 2 cm. (2012)

Question 19.

If a line segment intersects sides AB and AC of a ∆ABC at D and E respectively and is parallel to BC, prove that ADAB=AEAC. (2013)

Question 20.

In a ∆ABC, DE || BC with D on AB and E on AC. If ADDB=34 , find BCDE. (2013)

Question 21.

In the figure, if DE || OB and EF || BC, then prove that DF || OC. (2014)

Question 22.

If the perimeters of two similar triangles ABC and DEF are 50 cm and 70 cm respectively and one side of ∆ABC = 20 cm, then find the corresponding side of ∆DEF. (2014)

Question 23.

A vertical pole of length 8 m casts a shadow 6 cm long on the ground and at the same time a tower casts a shadow 30 m long. Find the height of tower. (2014)

Question 24.

In given figure, EB ⊥ AC, BG ⊥ AE and CF ⊥ AE (2015)

Prove that:

(a) ∆ABG ~ ∆DCB

(b) BCBD=BEBA

Question 25.

∆ABC ~ ∆PQR. AD is the median to BC and PM is the median to QR. Prove that ABPQ=ADPM. (2017D)

Question 26.

State whether the given pairs of triangles are similar or not. In case of similarity mention the criterion. (2015)

Question 29.

In the given figure, the line segment XY is parallel to the side AC of ∆ABC and it divides the triangle into two parts of equal areas. Find the ratio AXAB. (2017OD)

Question 30.

In the given figure, AD ⊥ BC and BD = 13CD. Prove that 2AC^{2} = 2AB^{2} + BC^{2}. (2012)

Question 31.

In the given figure, ∆ABC is right-angled at C and DE ⊥ AB. Prove that ∆ABC ~ ∆ADE and hence find the lengths of AE and DE. (2012, 2017D)

Question 32.

In ∆ABC, if AP ⊥ BC and AC^{2} = BC^{2} – AB^{2}, then prove that PA^{2} = PB × CP. (2015)

Question 33.

ABCD is a rhombus. Prove that AB^{2} + BC^{2} + CD^{2} + DA^{2} = AC^{2} + BD^{2}. (2013)

Question 34.

The diagonals of trapezium ABCD intersect each other at point o. If AB = 2CD, find the ratio of area of the ∆AOB to area of ∆COD. (2013)

Question 35.

The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO=CODO. Show that ABCD is a trapezium. (2014)

Question 36.

In a rectangle ABCD, E is middle point of AD. If AD = 40 m and AB = 48 m, then find EB. (2014D)

Question 37.

Let ABC be a triangle and D and E be two points on side AB such that AD = BE. If DP || BC and EQ || AC, then prove that PQ || AB. (2013)

Question 38.

In the figure, ∠BED = ∠BDE & E divides BC in the ratio 2 : 1.

Prove that AF × BE = 2 AD × CF. (2015)

Question 39.

In the given figure, AD = 3 cm, AE = 5 cm, BD = 4 cm, CE = 4 cm, CF = 2 cm, BF = 2.5 cm, then find the pair of parallel lines and hence their lengths. (2015)

Question 40.

If sides AB, BC and median AD of AABC are proportional to the corresponding sides PQ, QR and median PM of PQR, show that ∆ABC ~ ∆PQR. (2017OD)

Question 41.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. (2012)

Question 43.

In the given figure, BL and CM are medians of a triangle ABC, right angled at A. Prove that: 4(BL^{2} + CM^{2}) = 5BC^{2} (2012)

Question 44.

In the given figure, AD is median of ∆ABC and AE ⊥ BC. (2013)

Prove that b^{2} + c^{2} = 2p^{2} + 12 a^{2}.

Question 45.

In a ∆ABC, the perpendicular from A on the side BC of a AABC intersects BC at D such that DB = 3 CD. Prove that 2 AB^{2} = 2 AC^{2} + BC^{2}. (2013; 2017OD)

Question 46.

In ∆ABC, altitudes AD and CE intersect each other at the point P. Prove that: (2014)

(i) ∆APE ~ ∆CPD

(ii) AP × PD = CP × PE

(iii) ∆ADB ~ ∆CEB

(iv) AB × CE = BC × AD

Question 47.

In the figure, PQR and QST are two right triangles, right angled at R and T resepctively. Prove that QR × QS = QP × QT. (2014)

Question 48.

In the given figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that ar(ABC)ar(DBC)=AODO. (2012)

Question 49.

Hypotenuse of a right triangle is 25 cm and out of the remaining two sides, one is longer than the other by 5 cm. Find the lengths of the other

two sides. (2013)

Question 50.

In Figure, AB ⊥ BC, FG ⊥ BC and DE ⊥ AC. Prove that ∆ADE ~ ∆GCF. (2016 OD)