three triangles inscribed in a square. All tangents. What is the radius?
To find the radius of the inscribed circles in the three triangles inscribed in a square, we can follow these steps:
Step 1: Understand the Problem
In this problem, we have a square with three triangles inscribed in it. It is mentioned that the triangles are tangent to each other and tangent to the sides of the square. We need to determine the radius of the circle inscribed in each of these triangles.
Step 2: Recall Relevant Geometry Concepts
To find the radius of an inscribed circle in a triangle, we can use a formula that relates the radius, the area of the triangle, and the semiperimeter of the triangle. The formula is given by r = A / s, where r is the radius, A is the area of the triangle, and s is the semiperimeter of the triangle.
Step 3: Determine the Area and Semiperimeter of the Triangles
Since the triangles are inscribed in a square, their base is equal to the side length of the square. Let's call this side length "s". Therefore, each triangle has a base of "s".
The height of each triangle can be found by drawing perpendiculars from the center of the square to the sides of the square, forming three smaller triangles. These smaller triangles are similar to the larger triangles formed by the inscribed circles. By using similar triangles, we can find the height of the larger triangles in terms of the side length "s".
Step 4: Calculate the Area and Semiperimeter of the Triangles
Once we know the base and height of the triangles, we can calculate their areas.
Area of a triangle = 0.5 * base * height
Since the height is equal to the side length of the square minus twice the radius of the inscribed circle, we can substitute the values to find the area. Let's call the inscribed circle radius "r".
Area of a triangle = 0.5 * s * (s - 2r)
Since the triangles are all congruent, their areas will be the same.
Step 5: Find the Semiperimeter of the Triangles
The semiperimeter of a triangle is half the sum of its three sides. In our case, the sides consist of the base and the two other sides formed by the inscribed circle's radii.
Semiperimeter of a triangle = (base + side + side) / 2
For our triangles, the semiperimeter can be calculated as:
Semiperimeter of a triangle = (s + 2r + 2r) / 2 = (s + 4r) / 2
Step 6: Calculate the Radius
Now that we have the area and semiperimeter of the triangle, we can calculate the radius using the formula:
Radius = Area / Semiperimeter = (0.5 * s * (s - 2r)) / [(s + 4r) / 2]
To find the value of "r", we isolate it and solve the equation.
Step 7: Solve the Equation
Multiplying both sides of the equation by the denominator gives:
Radius * [(s + 4r) / 2] = 0.5 * s * (s - 2r)
Expanding and rearranging the equation gives:
(s + 4r) * Radius = s * (s - 2r)
Dividing both sides of the equation by "s" gives:
(1 + 4r/s) * Radius = s - 2r
Finally, isolate "r" by subtracting (1 + 4r/s) * Radius from both sides of the equation:
-4r/s * Radius = s - 2r - Radius
Simplifying the equation yields the final answer for "r".
Step 8: Solve for "r" by Simplifying the Equation
-s * 4r/s * Radius = s - 3r - Radius
-4r * Radius = s^2 - 3rs - s * Radius
r * (4 * Radius + s) = s^2 - 3rs
r = (s^2 - 3rs) / (4 * Radius + s)
Now, substitute the known values of "s" and "Radius" into the equation to calculate the final result.