Name: Illustration Based on the SAS Criterion
Uploaded: 2019-07-19
Duration: 4 min 36 s
Description: One of the easiest ways to learn a concept is through a visual illustration. To learn how the SAS congruence criterion is applied in mathematical problems, w...

Question

If the altitudes of a triangle are equal, prove that the triangle is equilateral.

Accepted Answer

Let $∆ A B C$ is a triangle and their altitudes $A D, B E a n d C F$ are equal.

We need to prove, $∆ A B C$ is an equilateral triangle.

We know that, altitudes are perpendicular to its base.

Now, in $∆ B E C and ∆ B F C$ ,

$B E = C F$ (Given)

$B C = B C$ (Common side)

$∠ B F C = ∠ B E C = 90 °$

So, $∆ B E C ≅ ∆ B F C$ (By R.H.S. congruency)

So, $∠ F B C = ∠ E C B$ -----(i) (Corresponding angles of congruent triangles are equal)

Now, in $∆ A D B and ∆ A D C$

$∠ A D B = ∠ A D C = 90 °$

$A D = A D$ (Common side)

$∠ F B C = ∠ E C B$ [from (i)]

So, $∆ A D B ≅ ∆ A D C$ (By A.S.A. congruency)

So, $A B = A C$ -----(ii) (Corresponding sides of congruent triangles)

Similarly, $∆ A C F ≅ ∆ B C F$ (By A.S.A. congruency)

So, $A C = B C$ -------(iii) (Corresponding sides of congruent triangles)

So, $A B = A C = B C$ [From equation (ii) and (iii)]

So, $∆ A B C$ is an equilateral triangle.

So, if the altitudes of a triangle are equal, prove that the triangle is equilateral.

Hence, proved.

Manipur Board Solutions for Chapter: Triangles, Exercise 2: EXERCISE 7.2

Manipur Board Mathematics Solutions for Exercise - Manipur Board Solutions for Chapter: Triangles, Exercise 2: EXERCISE 7.2

Questions from Manipur Board Solutions for Chapter: Triangles, Exercise 2: EXERCISE 7.2 with Hints & Solutions

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