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# geometry

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In triangle ABC, point X is on side BC such that  AX=13,BX=10,CX=10, and the circumcircles of triangles ABX and ACX have the same radius. Find the area of triangle ABC.

Mar 13, 2024

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Solving for the Area of Triangle ABC

We are given the following information:

Triangle ABC with point X on side BC such that AX = 13, BX = 10, and CX = 10.

The circumcircles of triangles ABX and ACX have the same radius (denoted by r).

1. Using Similar Triangles:

Since the circumcircles of triangles ABX and ACX have the same radius, they are similar. Furthermore, triangles ABX and ACX share angle A. Therefore, these triangles are also similar by AA Similarity.

This implies that the ratio between corresponding side lengths in these triangles is constant. In particular:

AX / AB = BX / AX (ratio of corresponding heights from X to the base)

2. Finding AB:

We are given that AX = 13 and BX = 10. Substituting these values into the equation above:

13 / AB = 10 / 13

Cross-multiplying:

AB * 10 = 13 * 13

Solving for AB:

AB = (13 * 13) / 10 = 16.9

3. Area of Triangle ABC:

Since we have the base (BC = BX + CX = 10 + 10 = 20) and the height (corresponding to base BC, which is the altitude drawn from A to BC), we can find the area of triangle ABC using the area formula:

Area of Triangle ABC = ½ * Base * Height

Height of Triangle ABC (altitude from A to BC) = AX (since triangles ABX and ACX are similar, the height from X in each triangle is proportional to the corresponding side length opposite X).

Area of Triangle ABC = ½ * 20 * 13 = 130

Therefore, the area of triangle ABC is 130 square units.

Mar 14, 2024