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State true or false.

In the given figure, $A B C$ is a triangle right-angled at $B$ such that $∠ B C A = 2 ∠ B A C$ , then $A C = 2 B C$ .

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True

False

answer is A.

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Detailed Solution

It is given that

A B C

is a triangle right-angled at

B

such that

∠ B C A = 2 ∠ B A C

.
Make construction on the given

Δ A B C

as shown in the figure given below.
Extend

C B

D

in such that

B D = B C

.
Join

A D

.
Assume

∠ C A B

and

∠ B A D

to be

x °

each.

Consider

Δ A B C

and

Δ A B D

B C = B D

(constructed)

∠ A B C = ∠ A B D

(each

90 °

)

A B = A B

(common)
Therefore, by SAS congruence criteria,

Δ A B C ≅ Δ A B D

By congruent parts of congruent triangles,

∠ C A B = ∠ D A B

A C = A D

......(1)
Calculate the measure of

∠ C A D

∠ C A D = ∠ C A B + ∠ B A D

∠ C A D = x ° + x °

∠ C A D = 2 x °

But,

A C = A D

from equation (1).

∠ A C D = ∠ A D B ⇒ 2 x °

Therefore,

Δ A C D

is an equilateral triangle.

A C = C D

A C = 2 B C

A B C

is a triangle right-angled at

B

such that

∠ B C A = 2 ∠ B A C

, then

A C = 2 B C

.
Hence the statement is true.
Hence option 1 is correct.

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