PYTHAGORAS THEOREM

1) The sides of certain triangles are given below. Determine which of them are right triangles :
a) a= 6cm, b= 8cm, and c= 10cm.        Y
b) a= 5cm, b= 8cm, and c= 11cm.        N
c) If the sides are3cm, 4cm and 6cm long.        N
d) a= 7cm, b= 24cm, and c= 25cm. Y
e) a= 9cm, b= 16cm, and c= 18cm.        N
f) a= 1.6cm, b= 10cm, and c= 6cm.        Y


2) A man goes 10m due to east and then 24m due north. Find the distance from the starting point.                           26m

3) A man goes 15 m due to west and then 8m due to North. How far is he from the starting point.                   17

4) A right angle triangle has hypotenuse of length p- q =1, find the length of the third side of the triangle.         √(2q+1)

5) In ∆ ABC, BC =12cm, CA=16cm and AB=20cm. Find the value of angle C.     

6) In ∆ ABC, AD perpendicular to BC, AD= BD=5cm, BC=17cm. Find the value of AC.

7) A ladder is placed in such a way that its foot is at a distance of 5m from a wall and its tip reachs a window 12 m above the ground. Determine the length of the ladder .                      13m

8) A ladder 25 m long reaches a window of a building 20m above the ground. Determine the distance of the foot of the ladder of the building.               15m

9) A ladder 15 long reaches a window which is 9m above the ground on one side of a street. Keeping its food at the same point, the ladder is turned to other side of the street to reach a window 12m high . Find the width of the street.    21m

10) The foot of the ladder is at a distance of 8m from the wall of a house and the top of the ladder is at a height of 15m from the ground. Find the length of the ladder.

11) A ladder 13m long reaches a window of a building 12 m above the ground. Determine the distance of the foot of the ladder from the building.

12) The foot of a ladder is 6m away from a wall and its top reaches a window 8m above the ground. If the ladder is shifted in such a way that its foot is 8m away from the wall, to what height does it's tip reach ?               6m

13) A ladder 17m long reaches a window of a building 15m above the ground. Find the distance of the foot of the ladder from the building.                       8m

14) The hypotenuse triangle is 6 m more than the twice of the shortest side. If the third side is 2 m less than hypotenuse , find the sides of the triangle.        10

15) In given figure, ABC is a right angle triangle, right angled at B. AD and CE are two medians drawn from A and C respectively. If AC=5cm and AD= 3√5/2 cm, 
find the length of CE.        2√5

16) ABC is a right angle triangle, right angled at C. Let BC=a, CA=b, AB= c and let p the length perpendicular from C on AB, find 
a) cp                        ab
b) 1/a²+ 1/b² .              1/p²

17) Three times the square of any side of an equilateral triangle is equal to ____ times the square of the altitude.
a) 2 b) 3 c) 4 d) 5

18) In an equilateral triangle with the side a than 
A) altitude is
a) a/2 b) √3/2 c) √3a d) √3a/2

B) Area is
a) a²/2 b) √3/4 c) √3a/4 d) √3a²/4

19) ABC is a right angle triangle, right angled at C and AC=√3 BC. Then angle ABC is
a) 30° b) 45° c) 60° d) 15°

20) ABC is a right angle triangle, right angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 6cm and 8cm. 
Find the radius of the circle.      2cm

21) If A be the area of right angle and b one of the sides containing the right angle, find the length of the altitude on the hypotenuse.       2Ab/√(b⁴+ 4A)

22) Two poles of heights 6m and 11m stand on a plane ground. If the distance between find their feet is 12m, find the distance between their tops.       13m

23) The heights of two vertical pillars are 7m and 12 m. The distance between their feet is 12 m. Find the distance between the tops.

24) Two poles of height 9m and 14m stand on a plane ground. If the distance between their feet is 12m, find the distance between the tops.


25) In an isosceles triangle ABC, AB= AC= 25cm, BC =14cm, calculate the altitude from A on BC. 24m

26) In an isosceles triangle ABC, if AB= AC =13cm and the altitude from A on BC is 5cm, find BC. 24cm

27) Determine the length of AD in terms of b and c shown in figure.      bc/√(b²+c²)

28) A triangle has sides 5cm, 12cm, and 13cm. Find the length of one decimal place, of the perpendicular from the opposite vertex to the side whose length is 13cm.    4.6

29) ABCD is a square. F is the mid-point of AB, BE is one third of BC. If the area of ∆FBE=108cm², find the length of AC.         50.904

30) In a ∆ ABC, AB= BC= CA= 2a and AD perpendicular to BC then find the value of
a) AD.                                   a√3
b) Area(∆ ABC).               √3 a²

31) The lengths of the diagonals of a rhombus are 24cm and 10cm. Find each side of the rhombus.          13cm

32) Each side of a rhombus is 10cm. If one of its diagonals is 16cm, find the length of other diagonal.      12cm

33) Calculate the height of an equilateral triangle each of whose sides measures 12cm.        10.39

34) In each of the figures given below, an altitude is drawn to the hypotenuse by a right- angled triangle. The length of different line segments are marked in each figure. Determine x,y,z in each case.      
a)
                                                 6,2√5,3√5

b) 
                                                 5,2√5,3√5

35) There is a staircase as shown in figure, connecting points A and B.
Measurements of steps are marked in the figure. Find the straight line distance between A and B.                            10

36) In the figure calculate 
A) YZ B) XZ C) XY.                         5,12,13

37) In the figure AB|| DC. M is the midpoint of AB. N is the midpoint of DC
Right angles and measurement of sides are indicated in the figure
calculate AC and AH.                   2, 2.09

38) From the diagram, Find
a) DE, AE.                            14.42,18.03
b) Find AD from the right angled triangle ADF.                   23.09

39) Given AB= a, BC= √2 a and AC=√3 a 
Find the angle ABC
a) 30 b) 45 c) 60 d) 90

40) Calculate
a) perimeter of quadrilateral WXZY.    66
b) Area of quadrilateral WXZY.          234

41) The diagonal of rhombus are 6cm and 8cm. Calculate the perimeter of the rhombus.           20cm

42) PQRS is a trapezium, PQ|| RS, PR meets PQ at 90°. Given PQ=9cm, PR=12cm, RS=5 cm. Calculate QR, PS, Area of PQRS.               15,13, 84

43) AB= AC= BC= 4cm, BC is produced to D such that CD=3 cm. 
Calculate AP and AD.                6.08, 3.46 cm

44) In the figure: angle PSR=90°, PQ=10cm, QS=6 cm, RQ= 9cm.
calculate the length of PR.               17cm

45) ∆ ABC is right angled triangle at B, AC=10cm. M is midpoint of BC. 
Work out the value of BC and CD, to nearest cm.                        8.0, 5.9583

46) Triangle ABC is a right angled at B; and BC= 6cm. PQBR is a rectangle. PQ= 2cm, PR= 8cm.
Calculate
a) value of angle PCR.                      63°26'
b) length of AC, correct to 1 dp.        13.4

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