I'm stuck on these two problems; can someone show me how to solve them step-by-step? Thanks
1.A baseball diamond is a square 90 ft on each side. How far is it diagonally from home plate to second base?
2.An equilateral triangle has sides 10 units long. Find (a)the height and (b) the area.
-MC
Do you know the Pythagorean theorem?
Yes; this is what my book says:
If the hypotenuse of a right triangle has length c, and the legs have lengths a and b, then c^2= a^2 + b^2
-MC
Ok, each of these cases involve a right triangle (second base;home;first base). You are looking for the hypotenuse c.
In the second, the vertical, half the base, and the side form the right triangle, you are looking for the vertical a.
Sure! Let's go through each problem step-by-step.
1. To find the distance diagonally from home plate to second base on a square baseball diamond, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the distance from home plate to second base forms the hypotenuse, and the sides of the square form the other two sides of the triangle.
Let's label the sides. Let's call the length of the side of the square "s". In this case, s = 90 ft. Then, the diagonal distance from home plate to second base is the hypotenuse, which we can call "d".
Using the Pythagorean Theorem equation, we have:
d^2 = s^2 + s^2
d^2 = 90^2 + 90^2
d^2 = 8100 + 8100
d^2 = 16200
To find the value of "d", we take the square root of both sides:
d = √16200
By calculating the square root, we find:
d ≈ 127.28 ft
Therefore, the diagonal distance from home plate to second base is approximately 127.28 ft.
2. To find the height and the area of an equilateral triangle, we can use the properties of the equilateral triangle.
(a) The height of an equilateral triangle can be found by drawing an altitude from one of the vertices (top corner) to the base (one of the sides) perpendicular to the base. Since all sides of an equilateral triangle are equal, the altitude will also be the height. It will divide the base into two equal parts and form a right triangle.
Using the Pythagorean Theorem, we can find the height (h):
h^2 = (side length)^2 - (half of the base)^2
h^2 = 10^2 - (10/2)^2
h^2 = 100 - 25
h^2 = 75
Taking the square root of both sides, we have:
h = √75
By calculating the square root, we find:
h ≈ 8.66 units
Therefore, the height of the equilateral triangle is approximately 8.66 units.
(b) The area of an equilateral triangle can be found by using the formula:
Area = (side length^2 * √3) / 4
Area = (10^2 * √3) / 4
Area = (100 * √3) / 4
Area = 25√3 / 2
Simplifying further, we can multiply the denominator and numerator by 2:
Area = 50√3 / 4
By simplifying the radical (√3), we can multiply it by its conjugate (√3):
Area = (50√3 * √3) / (4 * √3)
This simplifies to:
Area = 150√3 / 12
Further simplifying, we divide the numerator and denominator by 6:
Area = 25√3 / 2
Therefore, the area of the equilateral triangle is equal to 25√3 / 2 square units.