Question 1:
The Pythagorean theorem describes the fundamental relationship between the lengths of the sides in a right triangle. Specifically, it states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (called the legs or catheti). Mathematically, this relationship is expressed as:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\( c \)[/tex] is the hypotenuse, and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs of the triangle. This theorem helps in determining the length of one side when the lengths of the other two sides are known, making it a crucial tool for solving many geometric problems involving right triangles.
Question 2:
To find the distance from the ladder's base to the building, we can use the Pythagorean theorem. Let's solve the problem step by step:
1. Identify the known values:
- The length of the ladder (hypotenuse) is 10 feet.
- The height at which the ladder touches the building (one leg of the triangle) is 8 feet.
2. Apply the Pythagorean theorem:
The Pythagorean theorem formula is [tex]\( a^2 + b^2 = c^2 \)[/tex]. Here, [tex]\( c \)[/tex] is the hypotenuse, and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs. We are given [tex]\( c = 10 \)[/tex] feet and [tex]\( a = 8 \)[/tex] feet, and we need to find [tex]\( b \)[/tex].
Rearrange the formula to solve for [tex]\( b \)[/tex]:
[tex]\[ b^2 = c^2 - a^2 \][/tex]
3. Plug in the known values and solve for [tex]\( b \)[/tex]:
[tex]\[ b^2 = 10^2 - 8^2 \][/tex]
[tex]\[ b^2 = 100 - 64 \][/tex]
[tex]\[ b^2 = 36 \][/tex]
4. Calculate the length of the other leg ([tex]\( b \)[/tex]):
[tex]\[ b = \sqrt{36} \][/tex]
[tex]\[ b = 6 \][/tex]
Therefore, the distance from the base of the ladder to the building is 6 feet. This means the other leg of the triangle, which connects the base of the ladder to the building along the ground, measures 6 feet.
