In this problem, we have a triangle ABC with angles 75 degrees at A, 45 degrees at B, and 60 degrees at C. A semi-circle with diameter DE is drawn inside the triangle, where D is on side AB, E is on side AC, and the semi-circle touches side BC. We're told that BC equals 26 meters, and we need to find the radius of the semi-circle.
To solve this problem, we first need to find the height of triangle ABC. Given that angle B is 45 degrees and angle C is 60 degrees, angle A must be 75 degrees. Let's draw the altitude from A to BC, with F as the foot of the altitude. In right triangle ABF, angle B is 45 degrees, so angle BAF is also 45 degrees. This means BF equals AF, which is our height h. In right triangle ACF, angle C is 60 degrees, so angle CAF is 30 degrees. This means CF equals h divided by the square root of 3. Since BC equals BF plus CF, we can write: BC equals h plus h over square root of 3, which simplifies to h times the quantity 1 plus 1 over square root of 3. Given that BC equals 26 meters, we can solve for h, which equals 13 times the quantity 3 minus square root of 3.
Now, let's find the radius of the semi-circle. Let h1 be the altitude from A to the diameter DE. Since the semi-circle touches BC, the distance from DE to BC is equal to the radius R. Therefore, h1 equals h minus R. Since triangles ADE and ABC are similar, the ratio of corresponding altitudes equals the ratio of corresponding bases. So, h1 divided by h equals DE divided by BC. The diameter DE equals 2R, and BC equals 26, so this ratio equals R divided by 13. Substituting h1 equals h minus R, we get: (h minus R) divided by h equals R divided by 13. Solving this equation: 13 times (h minus R) equals R times h, which gives us 13h minus 13R equals Rh. Rearranging, we get 13h equals R times (h plus 13), so R equals 13h divided by (h plus 13). Substituting our value of h, which is 13 times (3 minus square root of 3), and simplifying, we get R equals 9 minus square root of 3.
After simplifying our expression, we find that the radius of the semi-circle is 9 minus square root of 3. Let's verify this answer numerically. The square root of 3 is approximately 1.732, so 9 minus square root of 3 is approximately 7.27 meters. Looking at our multiple-choice options, we can see that the correct answer is option 2: 9 minus square root of 3. This completes our solution to the problem.
To summarize what we've learned: When working with a triangle containing an inscribed semi-circle, we can use similar triangles to establish relationships between the radius and other dimensions. We first calculated the height of the triangle using trigonometric relationships, then used the similarity of triangles to find the radius. Breaking down complex geometric problems into clear steps makes them more manageable. Our final answer of 9 minus square root of 3, which is approximately 7.27 meters, combines concepts from triangle geometry, trigonometry, and circle properties. This approach of establishing relationships between different parts of a geometric figure is a powerful problem-solving technique.