A. how far is the library from the park?
b. how far is the park from the football field?

a. The distance from the library to the park is 2√5 miles

b. The distance from the park to the football field is 2√35 miles

The last answer ⇒ 2√5 miles , 2√35 miles

Step-by-step explanation:

* Lets revise the rules in the right angle triangle when we draw the

 perpendicular from the right angle to the hypotenuse

- In triangle ABC

# Angle B is a right angle

# The hypotenuse is AC

# BD ⊥ AC

∴ (AB)² = AD × AC

∴ (BC)² = CD × AC

∴ (BD)² = AD × CD

∴ BD × AC = AB × BC

* Lets use one of these rules to solve the problem

- Vertex B represents the position of the park

- Vertex A represents the position of the home

- Vertex C represents the position of the football field

- Vertex D represents the position of the library

- AB represents the distance from the home to the park

- CB represents the distance from the football field to the park

- DB represents the distance from the library to the park

- AC represents the distance from the home to the football field

- AD represents the distance from the home to the library

- CD represents the distance from the football field to the library

* Now lets solve the problem

a.

∵ The distance from the home to the library is 4 miles

∴ AD = 4

∵ The distance from the football field to the library is 10 miles

∴ CD = 10

∵ The distance from the library to the park represented by BD

- Lets use the rule (BD)² = AD × CD

∴ (BD)² = 4 × 10 = 40 ⇒ take √ for both sides

∴ BD = √40 = 2√5 miles

* The distance from the library to the park is 2√5 miles

b.

∵ The distance from the park to the football field represented by BC

* Lets use the rule (BC)² = CD × AC

∵ CD = 10

∵ AC = 4 + 10 = 14

∴ (BC)² = 10 × 14 = 140 ⇒ take √ for both sides

∴ BC = √140 = 2√35 miles

* The distance from the park to the football field is 2√35 miles