Question 4 (ii)
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that
AB = AC,
$i . e . Δ A B C i s a n i s o s c e l e s t r i a n g l e .$

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Solution

$I n Δ A E B a n d Δ A F C, W e h a v e ∠ A = ∠ A (C o m m o n a n g l e) B E = C F (G i v e n) ∠ B E A = ∠ C F A (E a c h e q u a l t o 90^{\circ}) T h e r e f o r e, b y A A S r u l e, Δ A E B ≅ Δ A F C$
$W e h a v e p r o v e d a b o v e t h a t Δ A E B ≅ Δ A F C . ∴ A B = A C (C o r r e s p o n d i n g p a r t s o f c o n g r u e n t t r i a n g l e s a r e e q u a l) \Rightarrow Δ A B C i s i s o s c e l e s .$

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Q. Question 4 (ii)
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that
AB = AC,

i . e . Δ A B C i s a n i s o s c e l e s t r i a n g l e .

$A B C$ is a triangle in which altitudes $B E$ and $C F$ to sides $A C$ and $A B$ are equal (see the given figure). Show that

(i) $△ A B E ≅ △ A C F$

(ii) $A B = A C,$ i.e., $A B C$ is an isosceles triangle.

Q. Question 3
ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal.

Q. Question 3
ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal.

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Question 4 (ii) ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that AB = AC, i.e. ΔABC is an isosceles triangle.

In ΔAEB and ΔAFC, We have∠ A=∠ A(Common angle)BE=CF(Given)∠ BEA=∠CFA (Each equal to 90∘)Therefore, by AAS rule,ΔAEB≅ΔAFC We have proved above that ΔAEB≅ΔAFC.∴AB=AC(Corresponding parts of congruent triangles are equal)⇒ΔABC is isosceles.

Question 4 (ii)
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that
AB = AC,
$i . e . Δ A B C i s a n i s o s c e l e s t r i a n g l e .$