SATHEE: Properties of Triangles 1 Question 16

Properties of Triangles 1 Question 16

16. There exists a $△ A B C$ satisfying the conditions

(a) $b \sin A = a, A < \frac{π}{2}$

(b) $b \sin A > a, A > \frac{π}{2}$

(d) $b \sin A < a, A < \frac{π}{2}, b > a$

Fill in the Blanks

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Answer:

Correct Answer: 16. (a, d)

Solution:

The sine formula is

$\frac{a}{\sin A} = \frac{b}{\sin B} \Rightarrow a \sin B = b \sin A$

(a) $b \sin A = a \Rightarrow a \sin B = a$

$\Rightarrow B = \frac{π}{2}$

Since, $∠ A < \frac{π}{2}$ , therefore the triangle is possible.

(b) and (c) $b \sin A > a$

$\Rightarrow a \sin B > a \Rightarrow \sin B > 1$

$∴ △ A B C$ is not possible.

(d) $b \sin A < a$

$\Rightarrow a \sin B < a \Rightarrow \sin B < 1 \Rightarrow ∠ B$ exists.

Now, $b > a \Rightarrow B > A$

Since, $A < \frac{π}{2}$

$∴$ The triangle is possible.

Hence, (a) and (d) are the correct answers.

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Properties of Triangles 1 Question 16

16. There exists a △ABC satisfying the conditions

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16. There exists a $△ A B C$ satisfying the conditions