Question

Classify the following triangles:

1. In ΔABC, AB² + BC² = AC². What type of triangle is ABC? 
2. The circumcentre of the triangle lies at the ‘midpoint of one side.
3. In ΔPQR, all altitudes are equal.
4. The circumcentre of the triangle lies outside the triangle.
   

Answer 1

Let's classify the triangles based on the given conditions:

i. In ΔABC, AB² + BC² = AC².

• This condition represents the Pythagorean theorem, which is true for a right-angled triangle. In a right-angled triangle, the square of the hypotenuse AC equals the sum of the squares of the other two sides AB and BC.
• Right-angled triangle.

ii. The circumcenter of the triangle lies at the midpoint of one side.

• The circumcenter is the point where the perpendicular bisectors of the sides of the triangle meet. If it lies at the midpoint of one side, the triangle must be a right-angled triangle since the circumcenter of a right triangle lies at the midpoint of the hypotenuse.
• Right-angled triangle.

iii. In ΔPQR, all altitudes are equal.

• If all the altitudes of a triangle are equal, the triangle must be equilateral. In an equilateral triangle, all sides are equal, and the altitudes perpendiculars from the vertices to the opposite sides are also equal.
• Equilateral triangle.

iv. The circumcenter of the triangle lies outside the triangle.

• The circumcenter of a triangle lies outside the triangle only in the case of an obtuse-angled triangle. In an obtuse-angled triangle, the circumcenter is located outside the triangle.
• Obtuse-angled triangle.