Question

14. Kathy lives directly east of the park. The football field is directly south of the park. The library (1 point) 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.

Park

Home

9 miles

Library

12 miles

Football field

a. How far is the library from the park?

b. How far is the park from the football field?

6.√3 miles; 6.7 miles

6.7 miles: 6.3 miles

√33 miles. √21 miles

√21 miles, √33 miles

Answer 1

(a) 6√3, 6√7

Explanation:

The right triangles shown are all similar, so corresponding sides are proportional. This gives rise to three "geometric mean" relations between the various lengths.

Geometric mean relations

Using P, F, L, and H to represent the points marked Park, Football field, Library, and Home, the relations are ...

PF/FH = FL/PF ⇒ PF = √(FH·FL)

PL/FL = HL/PL ⇒ PL = √(FL·HL)

PH/HL = HF/PH ⇒ PH = √(HL·HF)

Application

(a) The distance from the park to the library is ...

PL = √(FL·HL) = √(12·9) = 6√3 . . . miles

(b) The distance from the football field to the park is ...

PF = √(FH·FL) = √(21·12) = 6√7 . . . miles