In triangle ABC, point X is on side BC such that AX = 10, BC = 10, CX = 5, and the circumcircles of traingles ABX and ACX have the same radius. Find the area of triangle ABC.

 May 4, 2024

To find the area of triangle ABC, we can first determine the height of the triangle from point A to side BC. Let's denote the height as h.

1. Let's use the formula for the area of a triangle: Area = 1/2 x base x height. 


2. Since we know AX = 10, BC = 10, and CX = 5, we can find the length of BX by subtracting CX from BC: BX=BC-CX = 10-5=5.


3. Now, let's consider triangle ABX and triangle ACX. Given that the circumcircles of these triangles have the same radius, we can deduce that triangles ABX and ACX are similar triangles. 4. By the property of similar triangles, the ratio of the sides of the two triangles is equal: 5. Since AX = 10, CX = 5, and BC = 10, we can find AB using the ratio: AX/AB = CX/AC = BX/BC Solving for AB gives: AB=2xAX=2x10=20


6. Now that we have AB = 20 and BC = 10, we can use these side lengths to find the height of triangle ABC.

7. To find the height h, we can use the Pythagorean theorem: Solving for h gives:\(h=\sqrt{400-25}=\sqrt{375}=5\sqrt3\)


8. Finally, we can calculate the area of triangle ABC using the formula we mentioned earlier: Therefore, the area of triangle ABC is Area= 1/2 x BC x h= 1/2x10x\(5\sqrt3=25\sqrt3\)

 May 4, 2024

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