Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (-1, -1) are the vertices of a right angled triangle.

Question

A line passing through the point \( A(9, 0) \) makes an angle of \( 30^\circ \) with the positive direction of the x-axis. If this line is rotated about \( A \) through an angle of \( 15^\circ \) in the clockwise direction, then its equation in the new position is:

Answer 1

Given:

The three points are:
A(4, 4), B(3, 5) and C(−1, −1)

Step 1: Use slope method

If the product of slopes of two sides of a triangle is −1, then the sides are perpendicular and the triangle is right angled.

Step 2: Find slope of AB

Slope of AB =

(5 − 4) / (3 − 4)

= 1 / (−1)

= −1

Step 3: Find slope of AC

Slope of AC =

(−1 − 4) / (−1 − 4)

= −5 / −5

= 1

Step 4: Check perpendicularity

Slope AB × Slope AC = (−1) × (1)

= −1

So, AB ⟂ AC

Conclusion:

Since the sides AB and AC are perpendicular at point A, the triangle formed by the points (4, 4), (3, 5) and (−1, −1) is a right angled triangle, right angled at A.

Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (-1, -1) are the vertices of a right angled triangle.

Solution and Explanation

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