Kristen lives directly east of the park. the football field is directly south of the park. the library sits on the line formed between kristen’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 2 miles from her home. the football field is 8 miles from the library. how far is library from the park? how far is the park from the football field?

Imagine right triangle PHF, where P - park, H - home and F - football field, then PH, PF are legs and HF is hypotenuse . Denote point L to be library. You know that point L lies on the segment FH and FL=8, LH=2. Also you know that PL is an altitude to the hypotenuse.

Use the property of altitude drawn from the vertex of right angle to the hypotenuse (the length of the altitude is geometrical mean between legs' projections onto hypotenuse):

mi.

This means that the distance between park and libriry is 4 miles.

Consider right triangle PLF ( angle L is right angle and PF - hypotenuse). By the Pythagorean theorem,

mi. The distance between park and football field is miles.

the distance between park and libriry is 4 miles and the distance between park and football field is miles.

proportions are fractions that equal to each other. so if two fractions are the same, they are proportional.

let's check the answers to see if they are proportional.

question a is not proportional because you can't turn 1.5/3 to 3/9. if you try to multiply 1.5/3 by 2 or 3, you wont get 3/9. it's impossible to make the fractions the same.

question b is not proportional because you can't turn 22/33 to 4/6. 33 can't go into 6.

question c is proportional because when you multiply 2 and 3 by 71, you would get 142/213, and those fractions equal to each other

question d is proportional because when you divide 14 and 7 by 2, you would end up with 7/3.5, and those fractions equal to each other.