Page 186 - ICSE Math 8

Example 8: The diagonals of a rectangle DENT intersect at point O. D E
If –EON = 50º, find –ODT? 2
Solution: In rectangle DENT, diagonals DN and ET intersect at
point O, therefore, 1 O 50º
–1 = –EON (Vertically opposite angles)
3
fi –1 = 50º T N

Also, diagonal ET = diagonal DN (Diagonals of a rectangle are equal)
1
1
fi ET = OT = OE and DN = OD = ON
2 2
fi OD = OT
Now in D DOT, OD = OT (proved)
fi –2 = –3 (Angles opposite to equal sides are equal)

Also, –1 + –2 + –3 = 180º (Angle-sum property of a triangle)
fi 50º + –2 + –3 = 180º fi 2–2 = 180º – 50º = 130º
130∞
fi –2 = = 65º
2
Therefore, –2 = –ODT = 65º
Example 9: ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle.
Explain why O is equidistant from A, B and C.
Solution: Draw AD || BC and DC || AB to meet at D. Join OD. A D
Therefore, ABCD is a rectangle.
Since in a rectangle, diagonals are equal and they bisect
each other, so OA = OB = OC = OD. O
fi O is equidistant from A, B and C.

B C

EXERCISE 16.3

E C
1. PACE is a rectangle. Give reasons for the following:
(a) PC = AE (b) ∠APC = ∠CEA
(c) OA = OE = OP = OC (d) ∠POA = ∠COE O
2. In a square, one diagonal is 16 cm. Find the side of the square.
3. In a rectangle CALM, CA = 4 cm and AL = 3 cm. Find the P A
length of the diagonals CL and AM.
4. In a rectangle KITE, diagonal KT = 13 cm, IT = 5 cm. D C
Find KI, KE and ET.
5. In the figure given alongside, ABCD is a square. Find x. 85°
6. The sides of a rectangle are in the ratio 4 : 5. If the perimeter of the rectangle O
is 90 cm, find its sides. x
7. Name the quadrilaterals whose diagonals: A E B
(a) bisect each other.
(b) are perpendicular bisectors of each other.
(c) are equal.

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