Question 1 :
As observed from the top of a 75 m high lighthouse from the sea level, the Angles of depression of two ships are 30° and 45°. If one ship is exactly behind the ofher on the same side of the lighthouse, then find the distance between the two ships.
Question 2 :
A straight highway leads to the foof of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foof of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foof of the tower from this point.
Question 3 :
The angle of elevation of the top of a building from the foof of the tower is 30° and the angle of elevation of the top of the tower from the foof of building is 60°. If the tower is 50 m high, then find the height of the building.
Question 4 :
Two poless of equal heights are standing opposite to each ofher on either side of the road, which is 80 m wide. From a point between them on the road, the Angles of elevation of the top of the poless are 60° and 30°, respectively. Find the distances of the point from the poless.
Question 5 :
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foof of the tower, is 30°. Find the height of the tower.
Question 6 :
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eye to the top of the building increases from 30° to 60° as he walks tonwards the building. Find the distance he walked tonwards the building.
Question 7 :
From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are $45^\circ and 60^\circ$, respectively. Find the height of the tower.
Question 8 :
Two poless of equal heights are standing opposite to each ofher on either side of the road, which is 80 m wide. From a point between them on the road, the Angles of elevation of the top of the poless are 60° and 30°, respectively. Find the height of the poless.
Question 9 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19a61273b230584979932.jpeg' />
In the above image, a TV tower stands vertically on a bank of a canal. From a point on the ofher bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joining this point to the foof of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal.
Question 10 :
The Angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Find the height of the tower.
Question 11 :
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foof is 45°. Determine the height of the tower.
Question 12 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19a62273b230584979933.jpeg' />
In the above image, a 1.2 m tall girl spofs a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After sometime, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.
Question 13 :
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.
Question 14 :
A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.
Question 15 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19a60273b230584979931.jpeg' />
In the above image, a circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with ground level is 30°.
Question 16 :
A tree breaks due to storm and the broken part bends, so that the top of the tree touches the ground making an angle 30° with it. The distance between the foof of the tree to the point, where the top touches the ground is 8 m. Find the height of the tree.
Question 17 :
A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 yr, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m and inclined at an angle of 60° to the ground. What should be the length of the slides in each case?
Question 18 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c47273b230584979aaa.PNG' />
In the above fig, if PQ || RS, Is ∆ POQ ~ ∆ SOR?
Question 19 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c46273b230584979aa9.PNG' />
Are the two triangles given in the fig above similar ?
Question 20 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c4f273b230584979ab3.PNG' />
In the above fig, ∠ ACB = 90° and CD ⊥ AB. Is $\frac{BC^2}{AC^2} = \frac{BD}{AD}$ ?
Question 21 :
For going to a city B from city A, there is a route via city C such that AC is perpendicular to CB, AC = 2 x 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.
Question 22 :
Determine which of them are right triangles:
(i) 7 cm, 24 cm, 25 cm
(ii) 3 cm, 8 cm, 6 cm
(iii) 50 cm, 80 cm, 100 cm
(iv) 13 cm, 12 cm, 5 cm
Question 23 :
Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles find out if $\frac{OA}{OC}$ =$\frac{OB}{OD}$ ?
Question 24 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c6c273b230584979ad6.PNG' />
In the above fig, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥AC and OF ⊥AB. Is $AF^2 + BD^2 + CE^2$ = $AE^2 + CD^2 + BF^2$ ?
Question 25 :
ABC is an isosceles triangle right angled at C. Is$ AB^2$ = $2AC^2$ ?
Question 26 :
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. TRUE or FALSE?
Question 27 :
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.
Question 28 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c71273b230584979adc.PNG' />
In the above fig, AD is a median of a triangle ABC and AM ⊥ BC. Is $AB^2$ = $AD^2 – BC . DM + \left(\frac{BC}{2}\right)^2$ ?
Question 29 :
State True or False: The ratio of the corresponding altitudes of two similar triangles is $\frac{3}{5}$.Then it is correct to say that ratio of their areas is
$\frac{6}{5}$.
Question 30 :
CD and GH are respectively the bisectors of ∠ACB and ∠ EGF such that D and H lie on sides AB and FE of ∆ ABC and ∆ EFG respectively. If ∆ABC ~ ∆ FEG, is ∆ DCA ~ ∆ HGF ?
Question 31 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c45273b230584979aa7.PNG' />
In the above fig, $\frac{PS}{SQ}$ = $\frac{PT}{TR}$ and ∠ PST = ∠ PRQ. PQR is a/an __________ triangle.
Question 32 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c57273b230584979abd.PNG' />
In the above fig, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Is BC || QR ?
Question 33 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19b9d273b2305849799dd.png' />
If DE is parallel to BC, find the ratio of the area ADE and area DECB.
Question 34 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19ba1273b2305849799e2.png' />
In the above figure, ABC is a triangle right angled at B and BD is perpendicular to AC. If AD = 4 cm, and CD = 5 cm, find BD.
Question 35 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c50273b230584979ab5.PNG' />
In the above fig, BL and CM are medians of a triangle ABC right angled at A. $X(BL^2 + CM^2)$ = $Y (BC)^2$. What is the value of X and Y ?
Question 36 :
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.
Question 37 :
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 1.5 hours?
Question 38 :
CD and GH are respectively the bisectors of ∠ACB and ∠ EGF such that D and H lie on sides AB and FE of ∆ ABC and ∆ EFG respectively. If ∆ABC ~ ∆ FEG, is ∆ DCB ~ ∆ HGE ?
Question 39 :
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 3 m, find how far she is away from the base of the pole.
Question 40 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c70273b230584979adb.PNG' />
In the above fig, ABC is a triangle in which ∠ ABC < 90° and AD ⊥ BC. Is $AC^2$ =$AB^2 + BC^2 –2 BC . BD$ ?
Question 41 :
If in two triangles DEF and PQR, $\angle$D = $\angle$Q and $\angle$R = $\angle$E, then which of the following is not true?
Question 42 :
D is a point on the side BC of a triangle ABC such that ∠ADC = ∠ BAC. Is $CA^2= CB.CD$ true ?
Question 43 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c42273b230584979aa4.PNG' />
∆ ABC, DE || BC and $\frac{AD}{DB}$ = $\frac{AE}{EC}$. Which theorem is this ?
Question 44 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c68273b230584979ad1.PNG' />
In the above fig, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, is $\frac{ar (ABC)}{ar (DBC)}$ = $\frac{AO}{DO}$ ?
Question 45 :
State True or False: Two quadrilaterals are similar, if their corresponding angles are equal.
Question 46 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c66273b230584979acf.PNG' />
In the above fig, sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ∆ PQR. Is ∆ABC ~ ∆PQR ?
Question 47 :
State True or False: Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Then the two triangles will be similar.
Question 48 :
I - All congruent figures are similar.
II - All similar figures are congruent.
Which of these is correct ?
Question 49 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c56273b230584979abc.PNG' />
In the above fig, DE || OQ and DF || OR. Is EF || QR ?
Question 50 :
In triangles ABC and DEF, $\angle$B = $\angle$E, $\angle$F = $\angle$C and AB = 3 DE. Then, the two triangles are
Question 51 :
Areas of two similar triangles are 36 $cm^2$and 100 $cm^2$. If the length of a side of the larger triangle is 20 cm, find the length of the corresponding side of the smaller triangle.
Question 52 :
State True or False: It is given that $\Delta$ FED ~ $\Delta$ STU. Then $\frac{DE}{ST}=\frac{EF}{TU}$.
Question 54 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c61273b230584979ac9.PNG' />
In the above fig, altitudes AD and CE of ∆ ABC intersect each other at the point P. Is ∆AEP ~ ∆ CDP ?
Question 55 :
State True or False: It is given that $\Delta$ DEF ~ $\Delta$ RPQ. Then $\angle$D = $\angle$R and $\angle$F = $\angle$P.
Question 56 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c54273b230584979ab9.PNG' />
In the above fig (i) and (ii), DE || BC. Find AD in (ii).
Question 57 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c6e273b230584979ad8.PNG' />
In the above fig, PS is the bisector of ∠ QPR of ∆ PQR. Is $\frac{QS}{SR}$ = $\frac{PQ}{PR}$ ?
Question 58 :
State true or false:
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.
Question 59 :
State True or False: If in two right triangles, one of the acute angles of one triangle is equal to an acute angle of the other triangle, Then we can say that the two triangles will be similar.
Question 60 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c5e273b230584979ac5.PNG' />
In the above fig, ∆ ODC ~ ∆ OBA, ∠ BOC = 125° and ∠ CDO = 70°. Find ∠DCO.
Question 61 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19b9d273b2305849799dc.png' />
In the above figure, if AB is parallel to DC and AC and PQ intersect each other at the point O, then OA . CQ = OC . AP
Question 62 :
D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C.Is $AE^2 + BD^2$ = $AB^2 + DE^2$ ?
Question 63 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c64273b230584979acc.PNG' />
In the above fig, altitudes AD and CE of ∆ ABC intersect each other at the point P. Is ∆AEP ~ ∆ADB ?
Question 64 :
D and E are respectively the points on the sides AB and AC of a triangle ABC such that AD = 2 cm, BD = 3 cm, BC = 7.5 cm and DE is parallel to BC. Then, length of DE (in cm) is
Question 65 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c6b273b230584979ad5.PNG' />
In the above fig, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥AC and OF ⊥AB. Is $OA^2 + OB^2 + OC^2 – OD^2 – OE^2 – OF^2$ = $AF^2 + BD^2 + CE^2$ ?
Question 66 :
S and T are points on sides PR and QR of ∆ PQR such that ∠ P = ∠ RTS. Is ∆ RPQ ~ ∆ RST ?
Question 67 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c64273b230584979acd.PNG' />
In the above fig, ABC and AMP are two right triangles, right angled at B and M respectively. Is $\frac{CA}{PA}$ = $\frac{BC}{MP}$ ?
