Let ABC be a triangle in which AB = AC. Let L be the locus of points X inside or on the triangle such that BX = CX. Which of the following statements are correct? (1) L is a straight line passing through A and in-centre of triangle ABC is on L. (2) L is a straight line passing through A and orthocentre of triangle ABC is on L. (3) L is a straight line passing through A and centroid of triangle ABC is on L. Select the correct answer using the code given below.
1, 2 and 3
Understanding the Locus of Points X in an Isosceles Triangle
The question asks us to identify the locus of points X that are inside or on a triangle ABC, such that the distance from X to vertex B is equal to the distance from X to vertex C (i.e., BX = CX). We are given that the triangle ABC is isosceles with AB = AC.
The locus of points that are equidistant from two fixed points B and C is the perpendicular bisector of the line segment BC. In triangle ABC, since AB = AC, the triangle is isosceles with A as the apex. The altitude from A to the base BC, the median from A to BC, the angle bisector of angle A, and the perpendicular bisector of BC all coincide. This common line is the axis of symmetry for the isosceles triangle.
Therefore, the locus L of points X inside or on triangle ABC such that BX = CX is the segment of the perpendicular bisector of BC that lies inside or on the triangle. Since the perpendicular bisector of BC in an isosceles triangle ABC with AB = AC passes through vertex A, the locus L is a straight line segment passing through A and perpendicular to BC.
Analyzing Statements about the Locus L
Let's examine each statement given in the options.
- Statement 1: L is a straight line passing through A and in-centre of triangle ABC is on L.
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We know L is a straight line segment passing through A (part of the perpendicular bisector of BC).
The in-centre of a triangle is the intersection point of the angle bisectors. In an isosceles triangle ABC with AB = AC, the angle bisector of angle A is also the median from A, the altitude from A, and the perpendicular bisector of BC. Since L is the perpendicular bisector of BC passing through A, the angle bisector of angle A lies on L.
The in-centre always lies on the angle bisectors. Therefore, the in-centre of triangle ABC lies on the angle bisector of angle A, which in turn lies on L. So, the in-centre is on L.
Statement 1 is correct.
- Statement 2: L is a straight line passing through A and orthocentre of triangle ABC is on L.
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We know L is a straight line segment passing through A (part of the perpendicular bisector of BC).
The orthocentre of a triangle is the intersection point of the altitudes. In an isosceles triangle ABC with AB = AC, the altitude from A to BC is also the median from A, the angle bisector of angle A, and the perpendicular bisector of BC. Since L is the perpendicular bisector of BC passing through A, the altitude from A lies on L.
The orthocentre always lies on the altitudes. Therefore, the orthocentre of triangle ABC lies on the altitude from A, which in turn lies on L. So, the orthocentre is on L.
Statement 2 is correct.
- Statement 3: L is a straight line passing through A and centroid of triangle ABC is on L.
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We know L is a straight line segment passing through A (part of the perpendicular bisector of BC).
The centroid of a triangle is the intersection point of the medians. In an isosceles triangle ABC with AB = AC, the median from A to BC is also the altitude from A, the angle bisector of angle A, and the perpendicular bisector of BC. Since L is the perpendicular bisector of BC passing through A, the median from A lies on L.
The centroid always lies on the medians. Therefore, the centroid of triangle ABC lies on the median from A, which in turn lies on L. So, the centroid is on L.
Statement 3 is correct.
Conclusion
Based on our analysis, all three statements regarding the locus L (the segment of the perpendicular bisector of BC inside or on triangle ABC) are correct. The locus L is a straight line segment passing through vertex A, and it contains the in-centre, the orthocentre, and the centroid of the isosceles triangle ABC.
| Statement | Analysis | Correctness |
|---|---|---|
| L is a straight line passing through A and in-centre is on L. | L is the perpendicular bisector of BC through A. In an isosceles triangle, the angle bisector of A is on this line. In-centre is on the angle bisector. | Correct |
| L is a straight line passing through A and orthocentre is on L. | L is the perpendicular bisector of BC through A. In an isosceles triangle, the altitude from A is on this line. Orthocentre is on the altitude. | Correct |
| L is a straight line passing through A and centroid is on L. | L is the perpendicular bisector of BC through A. In an isosceles triangle, the median from A is on this line. Centroid is on the median. | Correct |
Revision Table: Triangle Centres in Isosceles Triangle
In an isosceles triangle ABC with AB=AC, the median, altitude, and angle bisector from vertex A to the base BC, as well as the perpendicular bisector of BC, all coincide. This line contains several key triangle centres:
| Triangle Centre | Definition | Location in Isosceles ▵ ABC (AB=AC) |
|---|---|---|
| In-centre | Intersection of angle bisectors | Lies on the angle bisector of ∠A, which is on the line L. |
| Orthocentre | Intersection of altitudes | Lies on the altitude from A, which is on the line L. |
| Centroid | Intersection of medians | Lies on the median from A, which is on the line L. |
Additional Information: Properties of Isosceles Triangles
An isosceles triangle is a triangle with two sides of equal length. The angles opposite these sides are also equal. The vertex where the two equal sides meet is called the apex vertex. The side opposite the apex is called the base.
Key properties related to the altitude, median, angle bisector from the apex, and perpendicular bisector of the base in an isosceles triangle (from vertex A to base BC where AB=AC):
- The altitude from A is perpendicular to BC.
- The median from A bisects BC.
- The angle bisector of ∠A divides ∠A into two equal angles.
- The perpendicular bisector of BC passes through the midpoint of BC and is perpendicular to BC.
In an isosceles triangle, these four lines from the apex to the base (altitude, median, angle bisector) and the perpendicular bisector of the base are all the same line. This line is an axis of symmetry for the triangle.
The locus of points equidistant from B and C is the perpendicular bisector of BC. In this specific isosceles triangle, this line passes through A and contains the in-centre, orthocentre, and centroid.
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