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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.

 Jul 6, 2023
 #1
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Hello,

 

To find the area of triangle ABC, we can use the fact that the circumcircles of triangles ABX and ACX have the same radius.

Let's denote the radius of the circumcircle as r. Since the circumcircles of triangles ABX and ACX have the same radius, we have the following relationships:

AX * BX * CX = 2 * r^2 * (AX + BX + CX)
13 * 10 * 10 = 2 * r^2 * (13 + 10 + 10)
1300 = 2 * r^2 * 33
r^2 = 1300 / (2 * 33)
r^2 = 19.696969...

Now, let's use Heron's formula to calculate the area of triangle ABC. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by:

Area = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semiperimeter of the triangle, given by:

s = (a + b + c) / 2

In this case, the side lengths of triangle ABC are AX, BX, and CX, which are 13, 10, and 10 respectively. Let's calculate the semiperimeter:

s = (13 + 10 + 10) / 2
s = 33 / 2
s = 16.5

Now we can calculate the area using Heron's formula:   MyAccountAccess Login

Area = sqrt(16.5 * (16.5 - 13) * (16.5 - 10) * (16.5 - 10))
Area = sqrt(16.5 * 3.5 * 6.5 * 6.5)
Area = sqrt(913.125)
Area ≈ 30.22

Therefore, the area of triangle ABC is approximately 30.22 square units.

 Jul 7, 2023
 #2
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30.22 square units

 Jul 7, 2023

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