SATHEE: Properties of Triangles 1 Question 15

Properties of Triangles 1 Question 15

15. Internal bisector of $∠ A$ of $△ A B C$ meets side $B C$ at $D$ . A line drawn through $D$ perpendicular to $A D$ intersects the side $A C$ at $E$ and side $A B$ at $F$ . If $a, b, c$ represent sides of $△ A B C$ , then

$(2006, 5$ M)

(a) $A E$ is HM of $b$ and $c$

(b) $A D = \frac{2 b c}{b + c} \cos \frac{A}{2}$

(d) $△ A E F$ is isosceles

Show Answer

Answer:

Correct Answer: 15. $n =$

Solution:

Since, $△ A B C = △ A B D + △ A C D$

$\Rightarrow \frac{1}{2} b c \sin A = \frac{1}{2} c A D \sin \frac{A}{2} + \frac{1}{2} b A D \sin \frac{A}{2}$

Again, $A E = A D \sec \frac{A}{2} = \frac{2 b c}{b + c}$

$\Rightarrow A E$ is HM of $b$ and $c$ .

$\begin{aligned} E F & = E D + D F = 2 D E = 2 A D \tan \frac{A}{2} \\ = 2 \frac{2 b c}{b + c} \cos \frac{A}{2} \tan \frac{A}{2} = \frac{4 b c}{b + c} \sin \frac{A}{2} \end{aligned}$

Since, $A D ⊥ E F$ and $D E = D F$ and $A D$ is bisector.

$\Rightarrow △ A E F$ is isosceles.

Hence, (a), (b), (c), (d) are correct answers.

पिछला

अगला

Properties of Triangles 1 Question 15

15. Internal bisector of $∠ A$ of $△ A B C$ meets side $B C$ at $D$ . A line drawn through $D$ perpendicular to $A D$ intersects the side $A C$ at $E$ and side $A B$ at $F$ . If $a, b, c$ represent sides of $△ A B C$ , then

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Properties of Triangles 1 Question 15

15. Internal bisector of ∠A of △ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and side AB at F. If a,b,c represent sides of △ABC, then

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Menu

15. Internal bisector of $∠ A$ of $△ A B C$ meets side $B C$ at $D$ . A line drawn through $D$ perpendicular to $A D$ intersects the side $A C$ at $E$ and side $A B$ at $F$ . If $a, b, c$ represent sides of $△ A B C$ , then