CLASS X : CHAPTER - 6 TRIANGLES

IMPORTANT FORMULAS & CONCEPTS

Similar Objects: All those objects which have the same shape but different sizes are called similar objects.
Two triangles are similar if:

their corresponding angles are equal (or)
their corresponding sides have lengths in the same ratio (or proportional)

Two triangles \( \Delta ABC \) and \( \Delta DEF \) are similar if:

\( \angle A = \angle D, \angle B = \angle E, \angle C = \angle F \)
\( \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \)

Basic Proportionality Theorem (Thales Theorem):
If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
If in a \( \Delta ABC \), a straight line DE parallel to BC, intersects AB at D and AC at E, then: \[ \frac{AB}{AD} = \frac{AC}{AE} \quad \text{and} \quad \frac{AB}{DB} = \frac{AC}{EC} \]

Converse of Basic Proportionality Theorem:
If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

Angle Bisector Theorem:
The internal (external) bisector of an angle of a triangle divides the opposite side internally (externally) in the ratio of the corresponding sides containing the angle.

Converse of Angle Bisector Theorem:
If a straight line through one vertex of a triangle divides the opposite side internally (externally) in the ratio of the other two sides, then the line bisects the angle internally (externally) at the vertex.

CRITERIA FOR SIMILARITY OF TRIANGLES

1. AAA (Angle-Angle-Angle) similarity criterion:
If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.
Remark: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar (AA similarity).

2. SSS (Side-Side-Side) similarity criterion:
In two triangles, if the sides of one triangle are proportional (in the same ratio) to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

3. SAS (Side-Angle-Side) similarity criterion:
If one angle of a triangle is equal to one angle of the other triangle and if the corresponding sides including these angles are proportional, then the two triangles are similar.

Areas of Similar Triangles:
The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

If \( \Delta ABC \sim \Delta EFG \), then: \[ \frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta EFG)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{BC}{FG}\right)^2 = \left(\frac{CA}{GE}\right)^2 \]

Properties of Similar Triangles:

If two triangles are similar, then the ratio of the corresponding sides is equal to the ratio of their corresponding altitudes.
If two triangles are similar, then the ratio of the corresponding sides is equal to the ratio of the corresponding perimeters.

Pythagoras Theorem (Baudhayan Theorem):
In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Converse of Pythagoras Theorem:
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 first side is a right angle.

MCQ WORKSHEET-I

If in triangle ABC and DEF, \( \frac{AB}{DE} = \frac{BC}{FD} \), then they will be similar when:
1. \( \angle B = \angle E \)
2. \( \angle A = \angle D \)
3. \( \angle B = \angle D \)
4. \( \angle A = \angle F \)
It is given that \( \Delta ABC \sim \Delta PQR \) with \( \frac{BC}{QR} = \frac{1}{3} \), then \( \frac{\text{ar}(ABC)}{\text{ar}(PQR)} \) is equal to:
1. 9
2. 3
3. \( \frac{1}{3} \)
4. \( \frac{1}{9} \)
In \( \Delta ABC \), \( DE \parallel BC \) and \( AD = 4 \) cm, \( AB = 9 \) cm, \( AC = 13.5 \) cm, then the value of EC is:
1. 6 cm
2. 7.5 cm
3. 9 cm
4. none of these
In figure \( DE \parallel BC \) then the value of AD is:

Explanation: Triangle ABC with line DE parallel to BC intersecting AB at D and AC at E. AE=1.8, EC=5.4, DB=7.2.
1. 2 cm
2. 2.4 cm
3. 3 cm
4. none of the above
ABC and BDE are two equilateral triangles such that \( BD = \frac{2}{3}BC \). The ratio of the areas of triangles ABC and BDE are:
1. 2 : 3
2. 3 : 2
3. 4 : 9
4. 9 : 4
A ladder is placed against a wall such that its foot is at distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. The length of the ladder is:
1. 6.5 m
2. 7.5 m
3. 8.5 m
4. 9.5 m
If the corresponding sides of two similar triangles are in the ratio 4 : 9, then the areas of these triangles are in the ratio is:
1. 2 : 3
2. 3 : 2
3. 81 : 16
4. 16 : 81
If \( \Delta ABC \sim \Delta PQR \), \( BC = 8 \) cm and \( QR = 6 \) cm, the ratio of the areas of \( \Delta ABC \) and \( \Delta PQR \) is:
1. 8 : 6
2. 6 : 8
3. 64 : 36
4. 9 : 16
If \( \Delta ABC \sim \Delta PQR \), area of \( \Delta ABC = 81 \text{ cm}^2 \), area of \( \Delta PQR = 144 \text{ cm}^2 \) and \( QR = 6 \) cm, then length of BC is:
1. 4 cm
2. 4.5 cm
3. 9 cm
4. 12 cm
Sides of triangles are given below. Which of these is a right triangle?
1. 7 cm, 5 cm, 24 cm
2. 34 cm, 30 cm, 16 cm
3. 4 cm, 3 cm, 7 cm
4. 8 cm, 12 cm, 14 cm
If a ladder 10 m long reaches a window 8 m above the ground, then the distance of the foot of the ladder from the base of the wall is:
1. 18 m
2. 8 m
3. 6 m
4. 4 m
A girl walks 200m towards East and then she walks 150m towards North. The distance of the girl from the starting point is:
1. 350 m
2. 250 m
3. 300 m
4. 225 m

MCQ WORKSHEET-II

In the given figure, if \( DE \parallel BC \), then x equals:

Explanation: Triangle ABC with DE parallel to BC. AD=2, DB=3, DE=4, BC=x.
1. 6 cm
2. 10 cm
3. 8 cm
4. 12.5 cm
All ____________ triangles are similar.
1. isosceles
2. equilateral
3. scalene
4. right angled
All circles are __________
1. congruent
2. similar
3. not similar
4. none of these
All squares are __________
1. congruent
2. similar
3. not similar
4. none of these
In the given fig \( DE \parallel BC \) then the value of EC is:

Explanation: Triangle ABC, DE parallel to BC. AD=1.5, DB=3, AE=1.
1. 1 cm
2. 2 cm
3. 3 cm
4. 4 cm
In the given below figure, the value of \( \angle P \) is:

Explanation: Two triangles ABC and PQR. AB=3.8, AC=3\sqrt{3}, BC=6, Angle A=80, Angle B=60. PQ=12, PR=6\sqrt{3}, QR=7.6.
1. 60°
2. 80°
3. 40°
4. 100°
A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, then the length of her shadow after 4 seconds.
1. 1.2 m
2. 1.6 m
3. 2 m
4. none of these
A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
1. 42 m
2. 48 m
3. 54 m
4. none of these
\( \Delta ABC \sim \Delta DEF \) and their areas be, respectively, 64 cm² and 121 cm². If EF = 15.4 cm, the value of BC is.
1. 11.2 cm
2. 15.4 cm
3. 6.4 cm
4. none of these
ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of the areas of triangles ABC and BDE is:
1. 2 : 1
2. 1 : 2
3. 4 : 1
4. 1 : 4
Areas of two similar triangles are in the ratio 4 : 9. Sides of these triangles are in the ratio:
1. 2 : 3
2. 4 : 9
3. 81 : 16
4. 16 : 81

MCQ WORKSHEET-III

In the following fig. \( XY \parallel QR \) and \( \frac{PX}{XQ} = \frac{PY}{YR} = \frac{1}{2} \), then
1. \( XY = QR \)
2. \( XY = \frac{1}{3}QR \)
3. \( XY^2 = QR^2 \)
4. \( XY = \frac{1}{2}QR \)
In the following fig \( QA \perp AB \) and \( PB \perp AB \), then AQ is:
1. 15 units
2. 8 units
3. 5 units
4. 9 units
The ratio of the areas of two similar triangles is equal to the:
1. ratio of their corresponding sides
2. ratio of their corresponding attitudes
3. ratio of the squares of their corresponding sides
4. ratio of the squares of their perimeter
The areas of two similar triangles are 144 cm² and 81 cm². If one median of the first triangle is 16 cm, length of corresponding median of the second triangle is:
1. 9 cm
2. 27 cm
3. 12 cm
4. 16 cm
In a right triangle ABC, in which \( \angle C = 90^\circ \) and \( CD \perp AB \). If \( BC = a, CA = b, AB = c \) and \( CD = p \).
1. \( \frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2} \)
2. \( \frac{1}{p^2} \neq \frac{1}{a^2} + \frac{1}{b^2} \)
3. \( \frac{1}{p^2} < \frac{1}{a^2} + \frac{1}{b^2} \)
4. \( \frac{1}{p^2} > \frac{1}{a^2} + \frac{1}{b^2} \)
Given Quad. ABCD ~ Quad PQRS then x is:
1. 13 units
2. 12 units
3. 6 units
4. 15 units
If \( \Delta ABC \sim \Delta DEF \), \( \text{ar}(\Delta DEF) = 100 \text{ cm}^2 \) and \( AB/DE = 1/2 \) then \( \text{ar}(\Delta ABC) \) is:
1. 50 cm²
2. 25 cm²
3. 4 cm²
4. None of the above.
If the three sides of a triangle are \( a, \sqrt{3}a, \sqrt{2}a \), then the measure of the angle opposite to the longest side is:
1. 45°
2. 30°
3. 60°
4. 90°
The similarity criterion used for the similarity of the given triangles shown in fig (iii) is:

Explanation: Triangle DEF with angles 70, 80 and Triangle PQR with angles 80, 30. (Note: 180-70-80=30, so angles are 70, 80, 30. PQR has 80, 30, so third is 70).
1. AAA
2. SSS
3. SAS
4. AA
The similarity criterion used for the similarity of the given triangles shown in fig (iv) is:

Explanation: Triangle ABC with sides 2, 2.5, 3. Triangle PQR with sides 4, 5, 6. Ratios are 2/4 = 2.5/5 = 3/6 = 1/2.
1. AAA
2. SSS
3. SAS
4. AA

MCQ WORKSHEET-IV

A vertical pole of length 20 m casts a shadow 10 m long on the ground and at the same time a tower casts a shadow 50 m long, then the height of the tower.
1. 100 m
2. 120 m
3. 25 m
4. none of these
The areas of two similar triangles are in the ratio 4 : 9. The corresponding sides of these triangles are in the ratio:
1. 2 : 3
2. 4 : 9
3. 81 : 16
4. 16 : 81
The areas of two similar triangles \( \Delta ABC \) and \( \Delta DEF \) are 144 cm² and 81 cm², respectively. If the longest side of larger \( \Delta ABC \) be 36 cm, then the longest side of the similar triangle \( \Delta DEF \) is:
1. 20 cm
2. 26 cm
3. 27 cm
4. 30 cm
The areas of two similar triangles are in respectively 9 cm² and 16 cm². The ratio of their corresponding sides is:
1. 2 : 3
2. 3 : 4
3. 4 : 3
4. 4 : 5
Two isosceles triangles have equal angles and their areas are in the ratio 16 : 25. The ratio of their corresponding heights is:
1. 3 : 2
2. 5 : 4
3. 5 : 7
4. 4 : 5
If \( \Delta ABC \) and \( \Delta DEF \) are similar such that \( 2AB = DE \) and \( BC = 8 \) cm, then EF =
1. 16 cm
2. 112 cm
3. 8 cm
4. 4 cm
XY is drawn parallel to the base BC of a \( \Delta ABC \) cutting AB at X and AC at Y. If \( AB = 4BX \) and \( YC = 2 \) cm, then AY =
1. 2 cm
2. 6 cm
3. 8 cm
4. 4 cm
Two poles of height 6 m and 11 m stand vertically upright on a plane ground. If the distance between their foot is 12 m, the distance between their tops is:
1. 14 m
2. 12 m
3. 13 m
4. 11 m
If D, E, F are midpoints of sides BC, CA and AB respectively of \( \Delta ABC \), then the ratio of the areas of triangles DEF and ABC is:
1. 2 : 3
2. 1 : 4
3. 1 : 2
4. 4 : 5
If \( \Delta ABC \) and \( \Delta DEF \) are two triangles such that \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = \frac{2}{5} \), then \( \frac{\text{ar}(ABC)}{\text{ar}(DEF)} = \)
1. 2 : 5
2. 4 : 25
3. 4 : 15
4. 8 : 125
In triangles ABC and DEF, \( \angle A = \angle E = 40^\circ \), \( AB : ED = AC : EF \) and \( \angle F = 65^\circ \), then \( \angle B = \)
1. 35°
2. 65°
3. 75°
4. 85°
If ABC and DEF are similar triangles such that \( \angle A = 47^\circ \) and \( \angle E = 83^\circ \), then \( \angle C = \)
1. 50°
2. 60°
3. 70°
4. 80°

PRACTICE QUESTIONS

State whether the following pairs of polygons are similar or not.

Explanation: A square and a rhombus with sides not proportional or angles not equal.
In triangle ABC, \( DE \parallel BC \) and \( \frac{AD}{DB} = \frac{3}{5} \). If \( AC = 4.8 \) cm, find AE.
A girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.
Diagonals of a trapezium ABCD with \( AB \parallel CD \) intersects at O. If \( AB = 2CD \), find the ratio of areas of triangles AOB and COD.
Prove that the areas of two similar triangles are in the ratio of squares of their corresponding altitudes.
In the figure, the line segment XY is parallel to side AC of \( \Delta ABC \) and it divides the triangle into two equal parts of equal areas. Find the ratio \( \frac{AX}{AB} \).

In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. Prove it.
E is a point on the side AD produced of a \( \parallel \text{gm} \) ABCD and BE intersects CD at F. Show that \( \Delta ABE \sim \Delta CFB \).
Complete the sentence: Two polygons of the same number of sides are similar if…….
In \( \Delta ABC \), \( AD \perp BC \). Prove that \( AB^2 - BD^2 = AC^2 - CD^2 \).
AD is a median of \( \Delta ABC \). The bisector of \( \angle ADB \) and \( \angle ADC \) meet AB and AC in E and F respectively. Prove that \( EF \parallel BC \).
State and prove the Basic Proportionality theorem. In the figure, if \( LM \parallel CB \) and \( LN \parallel CD \), prove that \( \frac{AM}{AB} = \frac{AN}{AD} \).
In the figure, \( DE \parallel BC \), find EC.
In the figure, \( DE \parallel BC \), find AD.
In given figure \( \frac{AD}{DB} = \frac{AE}{EC} \) and \( \angle AED = \angle ABC \). Show that \( AB = AC \).
\( \Delta ABC \sim \Delta DEF \), such that \( \text{ar}(\Delta ABC) = 64 \text{ cm}^2 \) and \( \text{ar}(\Delta DEF) = 121 \text{ cm}^2 \). If \( EF = 15.4 \) cm, find BC.
ABC and BDE are two equilateral triangles such that D is the midpoint of BC. What is the ratio of the areas of triangles ABC and BDE?
Sides of 2 similar triangles are in the ratio 4 : 9. What is the ratio areas of these triangles?
Sides of a triangle are 7cm, 24 cm, 25 cm. Will it form a right triangle? Why or why not?
Find \( \angle B \) in \( \Delta ABC \), if \( AB = 6\sqrt{3} \) cm, \( AC = 12 \) cm and \( BC = 6 \) cm.
Prove that “If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio”.
Prove that “If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.”
If a line intersects sides AB and AC of a \( \Delta ABC \) at D and E respectively and is parallel to BC, prove that \( \frac{AD}{AB} = \frac{AE}{AC} \).
ABCD is a trapezium in which \( AB \parallel DC \) and its diagonals intersect each other at the point O. Show that \( \frac{AO}{BO} = \frac{CO}{DO} \).
Prove that “If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.
Prove that “If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
Prove that “If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.
D is a point on the side BC of a triangle ABC such that \( \angle ADC = \angle BAC \). Show that \( CA^2 = CB \cdot CD \).
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of \( \Delta PQR \). Show that \( \Delta ABC \sim \Delta PQR \).
Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that \( \Delta ABC \sim \Delta PQR \).
If AD and PM are medians of triangles ABC and PQR, respectively where \( \Delta ABC \sim \Delta PQR \), prove that \( \frac{AB}{PQ} = \frac{AD}{PM} \).
A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
Prove that “The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.”
If the areas of two similar triangles are equal, prove that they are congruent.
D, E and F are respectively the mid-points of sides AB, BC and CA of \( \Delta ABC \). Find the ratio of the areas of \( \Delta DEF \) and \( \Delta ABC \).
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
Prove that “If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.”
Prove that “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
O is any point inside a rectangle ABCD. Prove that \( OB^2 + OD^2 = OA^2 + OC^2 \).
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
In Fig., if \( AD \perp BC \), prove that \( AB^2 + CD^2 = BD^2 + AC^2 \).
BL and CM are medians of a triangle ABC right angled at A. Prove that \( 4(BL^2 + CM^2) = 5 BC^2 \).
An aeroplane leaves an airport and flies due north at a speed of 1000 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 \( 1\frac{1}{2} \) hours?
D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that \( AE^2 + BD^2 = AB^2 + DE^2 \).
The perpendicular from A on side BC of a \( \Delta ABC \) intersects BC at D such that \( DB = 3 CD \). Prove that \( 2AB^2 = 2AC^2 + BC^2 \).
In an equilateral triangle ABC, D is a point on side BC such that \( BD = \frac{1}{3}BC \). Prove that \( 9AD^2 = 7AB^2 \).
P and Q are the points on the sides DE and DF of a triangle DEF such that \( DP = 5 \) cm, \( DE = 15 \) cm, \( DQ = 6 \) cm and \( QF = 18 \) cm. Is \( PQ \parallel EF \)? Give reasons for your answer.
Is the triangle with sides 25 cm, 5 cm and 24 cm a right triangle? Give reasons for your answer.
It is given that \( \Delta DEF \sim \Delta RPQ \). Is it true to say that \( \angle D = \angle R \) and \( \angle F = \angle P \)? Why?
A and B are respectively the points on the sides PQ and PR of a triangle PQR such that \( PQ = 12.5 \) cm, \( PA = 5 \) cm, \( BR = 6 \) cm and \( PB = 4 \) cm. Is \( AB \parallel QR \)? Give reasons for your answer.
In the Figure, BD and CE intersect each other at the point P. Is \( \Delta PBC \sim \Delta PDE \)? Why?
In triangles PQR and MST, \( \angle P = 55^\circ, \angle Q = 25^\circ, \angle M = 100^\circ \) and \( \angle S = 25^\circ \). Is \( \Delta QPR \sim \Delta TSM \)? Why?
Is the following statement true? Why? “Two quadrilaterals are similar, if their corresponding angles are equal”.
Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Are the two triangles similar? Why?
If in two right triangles, one of the acute angles of one triangle is equal to an acute angle of the other triangle, can you say that the two triangles will be similar? Why?
The ratio of the corresponding altitudes of two similar triangles is 3 : 5. Is it correct to say that ratio of their areas is 6 : 5 ? Why?
D is a point on side QR of \( \Delta PQR \) such that \( PD \perp QR \). Will it be correct to say that \( \Delta PQD \sim \Delta RPD \)? Why?
Is it true to say that if in two triangles, an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle, then the triangles are similar? Give reasons for your answer.
Legs (sides other than the hypotenuse) of a right triangle are of lengths 16cm and 8 cm. Find the length of the side of the largest square that can be inscribed in the triangle.
In the Figure, \( \angle D = \angle E \) and \( \frac{AD}{DB} = \frac{AE}{EC} \). Prove that BAC is an isosceles triangle.
Find the value of x for which \( DE \parallel AB \) in the figure.
In a \( \Delta PQR \), \( PR^2 - PQ^2 = QR^2 \) and M is a point on side PR such that \( QM \perp PR \). Prove that \( QM^2 = PM \times MR \).
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.
Diagonals of a trapezium PQRS intersect each other at the point O, \( PQ \parallel RS \) and \( PQ = 3 RS \). Find the ratio of the areas of triangles POQ and ROS.
Find the altitude of an equilateral triangle of side 8 cm.
If \( \Delta ABC \sim \Delta DEF \), \( AB = 4 \) cm, \( DE = 6 \) cm, \( EF = 9 \) cm and \( FD = 12 \) cm, find the perimeter of \( \Delta ABC \).
In the figure, if \( AB \parallel DC \) and AC and PQ intersect each other at the point O, prove that \( OA \cdot CQ = OC \cdot AP \).
In the figure, if \( DE \parallel BC \), find the ratio of ar (ADE) and ar (DECB).
ABCD is a trapezium in which \( AB \parallel DC \) and P and Q are points on AD and BC, respectively such that \( PQ \parallel DC \). If \( PD = 18 \) cm, \( BQ = 35 \) cm and \( QC = 15 \) cm, find AD.
Corresponding sides of two similar triangles are in the ratio of 2 : 3. If the area of the smaller triangle is 48 cm², find the area of the larger triangle.
In a triangle PQR, N is a point on PR such that \( QN \perp PR \). If \( PN \cdot NR = QN^2 \), prove that \( \angle PQR = 90^\circ \).
A 15 metres high tower casts a shadow 24 metres long at a certain time and at the same time, a telephone pole casts a shadow 16 metres long. Find the height of the telephone pole.
Areas of two similar triangles are 36 cm² and 100 cm². If the length of a side of the larger triangle is 20 cm, find the length of the corresponding side of the smaller triangle.
Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches.
An aeroplane leaves an Airport and flies due North at 300 km/h. At the same time, another aeroplane leaves the same Airport and flies due West at 400 km/h. How far apart the two aeroplanes would be after \( 1\frac{1}{2} \) hours?
It is given that \( \Delta ABC \sim \Delta EDF \) such that \( AB = 5 \) cm, \( AC = 7 \) cm, \( DF = 15 \) cm and \( DE = 12 \) cm. Find the lengths of the remaining sides of the triangles.
A 5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4 m high. If the foot of the ladder is moved 1.6 m towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.
In a triangle PQR, \( PD \perp QR \) such that D lies on QR. If \( PQ = a, PR = b, QD = c \) and \( DR = d \), prove that \( (a + b) (a - b) = (c + d) (c - d) \).
In the Figure, if \( \angle ACB = \angle CDA \), \( AC = 8 \) cm and \( AD = 3 \) cm, find BD.
In the Figure, if \( \angle 1 = \angle 2 \) and \( \Delta NSQ \cong \Delta MTR \), then prove that \( \Delta PTS \sim \Delta PRQ \).
In the Figure, OB is the perpendicular bisector of the line segment DE, \( FA \perp OB \) and FE intersects OB at the point C. Prove that \( \frac{1}{OA} + \frac{1}{OB} = \frac{2}{OC} \).
In the figure, line segment DF intersect the side AC of a triangle ABC at the point E such that E is the mid-point of CA and \( \angle AEF = \angle AFE \). Prove that \( \frac{BD}{CD} = \frac{BF}{CE} \).
In the figure, if \( \Delta ABC \sim \Delta DEF \) and their sides are of lengths (in cm) as marked along them, then find the lengths of the sides of each triangle.
In the figure, \( l \parallel m \) and line segments AB, CD and EF are concurrent at point P. Prove that \( \frac{AE}{BF} = \frac{AC}{BD} = \frac{CE}{FD} \).
In the figure, PQR is a right triangle right angled at Q and \( QS \perp PR \). If \( PQ = 6 \) cm and \( PS = 4 \) cm, find QS, RS and QR.
For going to a city B from city A, there is a route via city C such that \( AC \perp CB \), \( AC = 2x \) km and \( CB = 2(x + 7) \) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.
In the figure, ABC is a triangle right angled at B and \( BD \perp AC \). If \( AD = 4 \) cm and \( CD = 5 \) cm, find BD and AB.
In the figure, PA, QB, RC and SD are all perpendiculars to a line l, \( AB = 6 \) cm, \( BC = 9 \) cm, \( CD = 12 \) cm and \( SP = 36 \) cm. Find PQ, QR and RS.
In a quadrilateral ABCD, \( \angle A = \angle D = 90^\circ \). Prove that \( AC^2 + BD^2 = AD^2 + BC^2 + 2 \cdot CD \cdot AB \).
A flag pole 18 m high casts a shadow 9.6 m long. Find the distance of the top of the pole from the far end of the shadow.
A street light bulb is fixed on a pole 6 m above the level of the street. If a woman of height 1.5 m casts a shadow of 3m, find how far she is away from the base of the pole.
O is the point of intersection of the diagonals AC and BD of a trapezium ABCD with \( AB \parallel DC \). Through O, a line segment PQ is drawn parallel to AB meeting AD in P and BC in Q. Prove that PO = QO.
Prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle.
Prove that the area of the semicircle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the semicircles drawn on the other two sides of the triangle.
Using Thales theorem, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.
Using Converse of Thales theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.
In the figure A, B and C are points on OP, OQ and OR respectively such that \( AB \parallel PQ \) and \( AC \parallel PR \). Show that \( BC \parallel QR \).
In the figure, if \( \angle A = \angle C \), \( AB = 6 \) cm, \( BP = 15 \) cm, \( AP = 12 \) cm and \( CP = 4 \) cm, then find the lengths of PD and CD.
In the figure, if PQRS is a parallelogram, \( AB \parallel PS \) and \( PQ \parallel OC \), then prove that \( OC \parallel SR \).