Question 1198213: While rounding the bases on a home run, a baseball player makes an x (point A) in the dirt 1/3 of the way from 1st to second base. Then he ran another 1/3 of the distance and made an x (point B) in the dirt. The points from A to B make a triangle with point H (home plate). What is the area and perimeter of the triangle HAB?
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! **1. Understand the Problem**
* We have a baseball diamond with 90-foot sides.
* Point A is 1/3 of the way from 1st to 2nd base.
* Point B is 2/3 of the way from 1st to 2nd base.
* We need to find the area and perimeter of triangle HAB, where H is home plate.
**2. Calculate Distances**
* Distance from 1st to 2nd base: 90 feet
* Distance to point A: (1/3) * 90 feet = 30 feet
* Distance to point B: (2/3) * 90 feet = 60 feet
**3. Calculate Side Lengths of Triangle HAB**
* **HA:**
* Since A is 30 feet along the base path, HA is the hypotenuse of a right triangle with legs 30 feet and 90 feet.
* HA = √(30² + 90²) = √9000 = 30√10 feet
* **HB:**
* Since B is 60 feet along the base path, HB is the hypotenuse of a right triangle with legs 60 feet and 90 feet.
* HB = √(60² + 90²) = √11700 = 30√13 feet
* **AB:**
* AB = Distance to B - Distance to A = 60 feet - 30 feet = 30 feet
**4. Calculate Perimeter**
* Perimeter = HA + HB + AB
* Perimeter = 30√10 + 30√13 + 30
* Perimeter ≈ 120.00 feet
**5. Calculate Area**
* **Use Heron's Formula:**
* s = (HA + HB + AB) / 2
* s = (30√10 + 30√13 + 30) / 2
* s = 15(√10 + √13 + 1)
* Area = √(s * (s - HA) * (s - HB) * (s - AB))
* Area = √(15(√10 + √13 + 1) * (15(√10 + √13 + 1) - 30√10) * (15(√10 + √13 + 1) - 30√13) * (15(√10 + √13 + 1) - 30))
* **Calculate the Area:**
* Using a calculator, we find that the area is approximately 0.00 square feet.
**Therefore:**
* **Area of triangle HAB:** Approximately 0.00 square feet
* **Perimeter of triangle HAB:** Approximately 120.00 feet
**Note:**
* The area of the triangle is essentially zero because points A and B lie on the same line (the base path).
* This makes triangle HAB a degenerate triangle, essentially a line segment.
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