Heights and Distance – Assertion and Reason Questions

Q 1 :

Assertion (A): If the length of shadow of a tower is equal to its height, then the Sun’s altitude is $45 ° .$

Reason (R): $s i n 45 ° = 1 .$

Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
Assertion (A) is true but reason (R) is false.
Assertion (A) is false but reason (R) is true.

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(3)

Let AB be the tower and BC be its shadow on the ground such that $A B = B C = x$

$I n ∆ A B C, w e h a v e : t a n ∠ C = \frac{Perpendicular}{Base} = \frac{A B}{B C} \Rightarrow t a n ∠ C = \frac{x}{x} = 1 = t a n 45 ° \Rightarrow {S u n ’ s a l t i t u d e i s 45 \circ . H e n c e, A s s e r t i o n (A) i s t r u e . B u t R e a s o n (R) i s f a l s e b e c a u s e : s i n 45 ° = \frac{1}{\sqrt{2}} \neq 1$

Q 2 :

Assertion (A): The angle of elevation of top of a tower of height $20 \sqrt{3}$ m from a point on the ground, which is 60 metres away from the foot of the tower, is $30 ° .$

Reason (R): In a right triangle, $h^{2} = p^{2} + b^{2}$ , where h is hypotenuse, p is perpendicular (opposite side) and b is base (adjacent side) with respect to an acute angle.

Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
Assertion (A) is true but reason (R) is false
Assertion (A) is false but reason (R) is true.

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(2)

Let AB be the tower and C be a point on the ground.

$H e r e, A B = 20 \sqrt{3} m a n d B C = 60 m I n ∆ A B C, w e h a v e : t a n ∠ C = \frac{A B}{B C} = \frac{20 \sqrt{3}}{60} = \frac{1}{\sqrt{3}} \Rightarrow t a n ∠ C = t a n 30 ° \Rightarrow ∠ C = 30 °$

Hence, Assertion (A) is true.

Also, Reason (R) is true, as it states the Pythagoras theorem.
But it does not explain Assertion (A).

Q 3 :

Assertion (A): If the ratio of the height of a pole and its shadow is $\sqrt{3} : 1$ , then the Sun’s altitude is $30 °$ .

Reason (R): In a right triangle, $t a n θ = \frac{p}{b}$ , where p stands for perpendicular (opposite side to angle θ) and b for base (adjacent side).

Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
Assertion (A) is true but reason (R) is false.
Assertion (A) is false but reason (R) is true.

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(4)

Let AB be the pole and BC be its shadow on the ground.

Here,

$A B : B C = \sqrt{3} : 1 and let ∠ A C B = θ . I n ∆ A B C, w e h a v e : t a n θ = \frac{A B}{B C} = \frac{\sqrt{3}}{1} = \sqrt{3} \Rightarrow t a n θ = t a n 60 ° \Rightarrow θ = 60 ° \Rightarrow {S u n ’ s a l t i t u d e i s 60 \circ .$

Hence, Assertion (A) is false.

Also, the fact given under Reason (R) is the definition of tangent. So, Reason (R) is true.

Topic Question Set

Q 1 :

Assertion (A): If the length of shadow of a tower is equal to its height, then the Sun’s altitude is $45 ° .$

Reason (R): $s i n 45 ° = 1 .$

Q 3 :

Assertion (A): If the ratio of the height of a pole and its shadow is $\sqrt{3} : 1$ , then the Sun’s altitude is $30 °$ .

Reason (R): In a right triangle, $t a n θ = \frac{p}{b}$ , where p stands for perpendicular (opposite side to angle θ) and b for base (adjacent side).

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