Question

The distance between points A and P is the same as the distance between points B and P. If P does not lie on a line joining the points A and B, which of these conclusions is true?

a. Points A, B, and P form a straight line.
b. Points A, B, and P form an equilateral triangle.
c. Points A, B, and P form an isosceles triangle.
d. Points A, B, and P form a right-angled triangle.

Answer 1

Given the distances AP and BP are equal and point P does not lie on the line AB, points A, B, and P form an isosceles triangle, not a straight line, equilateral triangle, or right-angled triangle.

The correct conclusion is: c. Points A, B, and P form an isosceles triangle.

Step-by-step explanation:

The question is about determining the shape formed by points A, B, and P given that distances AP and BP are equal, and point P does not lie on the line joining A and B. Concluding that points A, B, and P form an isosceles triangle is correct. This is because in an isosceles triangle, two sides are equal in length and point P serves as the vertex where these equal sides (AP and BP) meet, while points A and B are at the ends of the base of the triangle.

Option a, stating that points A, B, and P form a straight line, cannot be correct because point P does not lie on the line joining A and B. Option b, suggesting an equilateral triangle, would imply all three distances between the points are equal, which is not given in the question. Option d, a right-angled triangle, would require a 90-degree angle at point P, which the information provided does not necessarily imply.