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CBSE Class 9 Mathematics Question 20 of 21

Triangles — Question 4

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Question 4

BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

Answer

Triangle ABC with BE and CF as equal altitudes is shown in the figure below:

BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles. NCERT Class 9 Mathematics CBSE Solutions.

Given :

BE is a altitude.

∴ ∠AEB = ∠CEB = 90°

CF is a altitude.

∴ ∠AFC = ∠BFC = 90°

Also, BE = CF.

In Δ BEC and Δ CFB,

⇒ ∠BEC = ∠CFB (Each equal to 90°)

⇒ BC = CB (Common)

⇒ BE = CF (Given)

⇒ Δ BEC ≅ Δ CFB (By R.H.S. congruence rule)

We know that,

Corresponding parts of congruent triangle are equal.

⇒ ∠BCE = ∠CBF (By C.P.C.T.)

As,

Sides opposite to equal angles of a triangle are equal.

∴ AB = AC.

Hence, proved that Δ ABC is an isosceles triangle.