Final answer:
To find the distance from the midpoint of the base to the opposite vertex in Karen's isosceles triangle field, we use the Pythagorean theorem and then convert feet to yards. The calculated distance is approximately 377.12 yards.
Step-by-step explanation:
Karen has a field in the shape of an isosceles triangle with two equal sides measuring 1,200 feet and a base of 800 feet.
To find the distance from the midpoint of the base to the opposite vertex, we must calculate the height of the triangle. Using the Pythagorean theorem, we can consider half of the base (400 feet) and the equal side (1,200 feet) to find the height. The formula is:
height = √(equal side length)2 - (half of the base)2
Substituting the values:
height = √(1,200 ft)2 - (400 ft)2 = √1,440,000 ft2 - 160,000 ft2 = √1,280,000 ft2 = 1,131.37 ft
To convert feet to yards, we divide by 3 (since 1 yard = 3 feet), so:
Distance in yards = 1,131.37 ft ÷ 3 ft/yd = 377.12 yd
Therefore, the distance from the midpoint of the triangle's base to the vertex, in yards, is approximately 377.12 yards.