Page 80 - Mathematics Class - XII

OBSERVATION
1. By actual measurement (Fig. (a)), we have:

OA 10 cm, OP 10 cm, OB 10 cm

AP 12 cm, BP 16 cm,, AB 20 cm

2 2 2
AP BP 12 2 16 2 144 256 400 20 2 AB

APB 90 [Pythagoras theorem]

AP . BP AP BP cos 90 0

2. Similarly by actual measurement (Fig. (b))

We have

OQ OA OB OP 10 cm

AQ 17 cm, BQ 10 5. cm

2 2
A AQ BQ 17 10 5 . 2 289 110 25 399 25 400 . . 400 approx..
2
AQB 90 [Pythagoras theorem]

Similarly, if we takes points R, S, T on semi-circle, we get
∠ARB = 90°, ∠ASB = 90°, ∠ATB = 90°

Hence, angle in a semi-circle is a right angle.

CONCLUSION
From this activity, it is verified that angle in a semi-circle is a right angle.

APPLICATION
This activity can be used to explain the concepts of

(i) opposite vectors
(ii) vectors of equal magnitude
(iii) perpendicular vectors
(iv) Dot product of two vectors.

Knowledge Booster
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● ab . is a real number.  



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● If a and are two non-zero vectors, then ab. = 0, if and only if a and are perpendicular to each other.
b
b
   
ab. 0 a b
    2
● If 0,then ab ab . In particular, aa. = a , as in this case, q is 0.
.
    2

● If ,then ab ab . In particular, a. a as in this case, q is π.
a
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