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Properties of Isosceles and Equilateral Triangles

In the given ...

Question

In the given figure, $△ A B C$ is an isosceles triangle in which $A B = A C .$ If $B D ⊥ A C$ and $C E ⊥ A B$ , prove that $B D = C E .$

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Solution

Given:
$Δ A B C$ is isosceles triangle.
and, $A B = A C$
$B D ⊥ A C & C E ⊥ A B$
To prove:
$B D = C E$
Proof:

In $Δ B E C & Δ C D B$ , we have
$∠ B E C = ∠ C D B = 90^{\circ}$ [ $B D$ and $C D$ are altitudes]
$∠ E B C = ∠ D C B$ [ $∵ A B = A C,$ angles opposite to equal sides are equal]
$B C = C B$ [Common in both triangles]
$Δ B E C ≅ Δ C D B$ ( by AAS congruence Rule)
Hence, $B D = C E$ (by CPCT)

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Similar questions

In the given figure, $△ A B C$ is an isosceles triangle in which $A B = A C$ . If $A B$ and $A C$ are produced to $D$ and $E$ respectively such that $B D = C E$ .
Prove that $B E = C D .$

Q. In given figure, If

B D ⊥ A C

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(i)

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Δ A B C

is an isosceles

Δ

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A B = A C

. If

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and

C E

are two medians of a triangle. Prove that

B D = C E

ABC

is a triangle in which altitudes

BD

and

CE

to sides

AC

and

AB

are equal (see figure). Show that

AB = AC

i.e.

ABC

is an isosceles triangle

Suppose ABC is an isosceles triangle with AB = AC; BD and CE are bisectors of ∠B and ∠C. Prove that BD = CE.

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In the given figure, △ABC is an isosceles triangle in which AB=AC. If BD⊥AC and CE⊥AB, prove that BD=CE.

In the given figure, $△ A B C$ is an isosceles triangle in which $A B = A C .$ If $B D ⊥ A C$ and $C E ⊥ A B$ , prove that $B D = C E .$