Chapter 6 The Triangles and its Properties Ex 6.5 Solutions
Question - 1 : - PQR is a triangle, right angled at P. If PQ = 10 cm and PR = 24 cm, find QR.
Answer - 1 : -
Inright angled triangle PQR, we have
QR2 = PQ2 +PR2 From Pythagoras property)
= (10)2 + (24)2
= 100 + 576 = 676
∴ QR =
= 26 cm
The, the required length of QR = 26 cm.
Question - 2 : - ABC is a triangle, right angled at C. If AB = 25 cm and AC = 7 cm, find BC.
Answer - 2 : -
In right angled ∆ABC, we have
BC2 + (7)2 =(25)2 (By Pythagoras property)
⇒ BC2 + 49 = 625
⇒ BC2 = 625 – 49
⇒ BC2 = 576
∴ BC =
= 24 cm
Thus, the required length of BC = 24 cm.
Question - 3 : - A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.
Answer - 3 : -
Here,the ladder forms a right angled triangle.
∴ a2 + (12)2 =(15)2 (By Pythagoras property)
⇒ a2+ 144 = 225
⇒ a2 =225 – 144
⇒ a2 = 81
∴ a =
= 9 m
Thus, the distance of the foot from the ladder = 9m
Question - 4 : - Which of the following can be the sides of a right triangle?
(i) 2.5 cm, 6.5 cm, 6 cm.
(ii) 2 cm, 2 cm, 5 cm.
(iii) 1.5 cm, 2 cm, 2.5 cm
Answer - 4 : -
(i) Given sides are 2.5 cm, 6.5 cm, 6 cm.
Square of the longer side = (6.5)2 =42.25 cm.
Sum of the square of other two sides
= (2.5)2 + (6)2 = 6.25 + 36
= 42.25 cm.
Since, the square of the longer side in a triangle is equal to the sum of thesquares of other two sides.
∴ Thegiven sides form a right triangle.
(ii) Given sides are 2 cm, 2 cm, 5 cm .
Square of the longer side = (5)2 =25 cm Sum of the square of other two sides
= (2)2 + (2)2 =4 + 4 = 8 cm
Since 25 cm ≠ 8 cm
∴ Thegiven sides do not form a right triangle.
(iii) Given sides are 1.5 cm, 2 cm, 2.5 cm
Square of the longer side = (2.5)2 =6.25 cm Sum of the square of other two sides
= (1.5)2 + (2)2 = 2.25 + 4
Since 6.25 cm = 6.25 cm = 6.25 cm
Since the square of longer side in a triangle is equal to the sum of square ofother two sides.
∴ Thegiven sides form a right triangle.
Question - 5 : - A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree . Find the original height of the tree.
Answer - 5 : -
Let ABbe the original height of the tree and broken at C touching the ground at Dsuch that
![](data:image/png;base64,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)
AC = 5 m
and AD = 12 m
In right triangle ∆CAD,
AD2 + AC2 =CD2 (By Pythagoras property)
⇒ (12)2 + (5)2 =CD2
⇒ 144 +25 = CD2
⇒ 169 =CD2
∴ CD =
= 13 m
But CD = BC
AC + CB = AB
5 m + 13 m = AB
∴ AB =18 m .
Thus, the original height of the tree = 18 m.
Question - 6 : - Angles Q and R of a APQR are 25° and 65°. Write whichof the following is true.
(i) PQ2 + QR2 = RP2
(ii) PQ2 + RP2 = QR2
(iii) RP2 + QR2 = PQ2
Answer - 6 : -
We know that
∠P + ∠Q + ∠R = 180° (Angle sum property)
∠P + 25°+ 65° = 180°
∠P + 90°= 180°
∠P =180° – 90° – 90°
∆PQR is a right triangle, right angled at P
(i) Not True
∴ PQ2 + QR2 ≠RP2 (By Pythagoras property)
(ii) True
∴ PQ2 + RP2 =QP2 (By Pythagoras property)
(iii) Not True
∴ RP2 + QR2 ≠PQ2 (By Pythagoras property)
Question - 7 : - Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.
Answer - 7 : -
Given:Length AB = 40 cm
Diagonal AC = 41 cm
In right triangle ABC, we have
AB2 + BC2=AC2 (By Pythagoras property)
⇒ (40)2 + BC2 =(41)2
⇒ 1600 +BC2 = 1681
⇒ BC2 = 1681 – 1600
⇒ BC2 = 81
∴ BC =
= 9 cm
∴ AB =DC = 40 cm and BC = AD = 9 cm (Property of rectangle)
∴ Therequired perimeter
= AB + BC + CD + DA
= (40 + 9 + 40 + 9) cm
= 98 cm
Question - 8 : - The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.
Answer - 8 : -
LetABCD be a rhombus whose diagonals intersect each other at O such that AC = 16cm and BD = 30 cm
Since, the diagonals of a rhombus bisect each other at 90°.
∴ OA =OC = 8 cm and OB = OD = 15 cm
In right ∆OAB,
AB2 = OA2 +OB2 (By Pythagoras property)
= (8)2+ (15)2 =64 + 225
= 289
∴ AB =
= 17 cm
Since AB = BC = CD = DA (Property of rhombus)
∴Required perimeter of rhombus
= 4 × side = 4 × 17 = 68 cm.