Question

If x2 + y2 + z2 = xy + yz + zx, then the triangle is:

Answer 1

The answer is in the attachment....

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Answer 2

If x² + y² + z² = xy + yz + zx then the triangle is equilateral triangle.

Given:

x² + y² + z² = xy + yz + zx

To Find:

The type of the triangle

Solution:

Step 1 of 3 :

Write down the given equation

Here the given equation is

x² + y² + z² = xy + yz + zx

Step 2 of 3 :

Simplify the given equation

x² + y² + z² = xy + yz + zx

⇒ 2x² + 2y² + 2z² = 2xy + 2yz + 2zx

⇒ 2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 0

⇒ x² - 2xy + y² + y² - 2yz + z² + z² - 2zx + x² = 0

⇒ (x - y)² + (y - z)²+ (z - x)² = 0

Step 3 of 3 :

State the type of the triangle

(x - y)² + (y - z)²+ (z - x)² = 0

We know that if sum of squares of three Real Numbers are zero then they are separately zero

Thus we get

x - y = 0 , y - z = 0 , z - x = 0

⇒ x = y , y = z , z = x

⇒ x = y = z

So three sides of the triangle are equal

Hence the triangle is an equilateral triangle.

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