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From the options given below, find which of the following statements are true.

(I) In a $Δ A B C$ in which $A B = A C$ , the altitude $A D$ bisects $B C$ .

(II) If the altitudes $A D, B E$ and $C F$ of $Δ A B C$ are equal, then $Δ A B C$ is equilateral.

(III) If $D$ is the midpoint of the hypotenuse $A C$ of a right $Δ A B C$ , then $B D = A C$ .

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I only

II only

I and II

II and III

answer is C.

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Detailed Solution

Given that, there are three statements.
For the statement (I),
Draw a rough diagram.
Question Image

Δ A B C

A B = A C

.
The altitude

A D

bisects

B C

.
Hence, the statement (I) is true.
For the statement (II),
Draw a rough diagram.
Question Image

The altitudes

A D, B E

and

C F

Δ A B C

are equal.
Hence,

Δ A B C

is equilateral.
Hence, statement (II) is true.
For statement (III),
Draw a rough diagram.
Question Image

A C

is the largest side of

Δ A B C

.
That is

A C > A B, B C

B D

is the altitude of

Δ A B C

.
In a right-angled triangle, the hypotenuse is

22

times the altitude.
Hence,

A C = 22 B D

.
Hence the statement is false.
Hence only statement (I), (II) is true.
Hence option 3 is correct.

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