Let's designate the vertices of the triangle with \(H\) as Kathy's home (top right), \(F\) as the football field (bottom left), and \(P\) as the park (upper left). The library \(L\) is where the altitude from \(P\) intersects the hypotenuse \(HF\).
Given:
- \(HL = 9\) miles
- \(LF = 12\) miles
- \(L\) is at the altitude drawn from \(P\) to \(HF\) and forms right angles with \(HF\).
First, we need to verify the relationship within the triangle and find the distance between the library (L) and the park (P).
1. Let’s denote:
- The distance \(PH = a\)
- The distance \(PF = b\)
By the definition of the altitude and the properties of right triangles:
\[ a^2 + b^2 = c^2 \]
\[ c = HL + LF = 9 + 12 = 21 \]
Given this and using the properties of the right triangle:
\[ x = PL \]
We know that
\[ (x^2 + 9^2) + (x^2 + 12^2) = 21^2 \]
Thus,
\[ x^2 + 81 + x^2 + 144 = 441 \]
\[ 2x^2 + 225 = 441 \]
\[ 2x^2 = 216 \]
\[ x^2 = 108 \]
\[ x = \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \]
So, the distance between Park (P) and Library (L) is \(6\sqrt{3}\).
Now, finding distance \(PF = b\) (Park to Football Field):
We consider the fact that (a^2 + b^2 = c^2)
We know:
\[ H = 6\sqrt{3}; 12; 21 \]
After calculation:
\[ 6 \sqrt 21 miles; 6 \sqrt 33 miles\]
Kathy lives directly east of the park. The football field is directly south of the park. The library sits on the line formed between Kathy’s home and the football field at the exact point where an altitude to the right triangle formed by her home, the park, and the football field could be drawn. The library is 9 miles from her home. The football field is 12 miles from the library.
The vertex at the lower left is labeled 'Football field.'
The vertex at the upper right is labeled 'Home.'
The vertex at the upper left is labeled 'Park' and is marked with a small square.
A line segment connects the upper left vertex to the hypotenuse at a point that is labeled 'Library.' The angle formed by the hypotenuse and the line segment is marked with a small square.
The distance between the football field and library is labeled 12 miles.
The distance between the library and home is labeled 9 miles.
How far is the library from the park?
How far is the park from the football field?
(1 point)
Responses
miles; miles
Image with alt text: 6 square root 3 miles; Image with alt text: 6 square root 7 miles
miles; miles
Image with alt text: 6 square root 7 miles; Image with alt text: 6 square root 3 miles
miles, miles
Image with alt text: square root 33 miles, Image with alt text: square root 21 miles
miles, miles
image with alt text: square root 21 miles, image with alt text: square root 33 miles
1 answer
1 answer