Page 182 - ICSE Math 7

Solution: Applying Pythagoras property, we get

2
2
2
2
(a) x + 8 = 10 (b) x 2 + 6 = 10 2
2
2
2
⇒ x = 10 – 8 2 ⇒ x 2 = 100 – 36 = 64 = 8 2
2
x
2
⇒ x = 36 ⇒ = 8
2
⇒ x = 6 cm ⇒ x = 16 cm
Example 6: Which of the following can be the sides of a right-angled triangle?
(a) 1.5 cm, 2.5 cm, 2 cm (b) 3 cm, 3 cm, 5 cm
In case of right-angled triangles, identify the right angle.
Solution: In each case we verify whether or not
2
2
(Hypotenuse) = (Base) + (Perpendicular) 2
2
2
(a) 1.5 + 2 = 2.25 + 4 = 6.25
2
Also, 2.5 = 6.25
2
2
Thus, 1.5 + 2 = 2.25 2
Hence, these sides form a right-angled triangle and the angle opposite the side
measuring 2.5 cm is a right angle.
2
2
2
(b) 3 + 3 = 9 + 9 = 18 ≠ 5 . Hence, these sides do not form a right-angled triangle.
Example 7: 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.
Solution: The broken tree forms ∆ ABC in which, D
2
2
CA = AB + BC 2
2
2
2
⇒ CA = 12 + 5 = 169
⇒ CA = 13 m, which is the upper part of the tree C
∴ Original height of the tree = BC + CD = BC + CA 5 m
= 5 m + 13 m = 18 m
A 12 m B
Example 8: A box is 8 cm long and 6 cm wide. If the height of the box is 24 cm, find the length of
the longest stick l which can come completely inside the box. H G
Solution: In ∆ DCB,
2
2
2
2
2
BD = DC + CB = 8 + 6 (Pythagoras property) E F
2
⇒ BD = 100 l 24 cm
⇒ BD = 10 cm
In ∆ HDB,
2
2
2
2
HB = HD + DB = 24 + 10 2 (Pythagoras property) D
C
2
⇒ l = 576 + 100 = 676 6 cm
⇒ l = 26 cm A 8 cm B
Try These
1. Find the perimeter and area of the rectangle whose breadth is 8 cm and a diagonal is 17 cm.
2. The diagonals of a rhombus measure 10 cm and 24 cm. Find its perimeter.

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