Problem:
Point B is due east of point A. Point C is due north of point B. The distance between points A and C is 102​ meters, and ∠BAC=45∘. Point D is 20 meters due north of point C. The distance AD is between which two integers?
Answer Choices:
A. 30 and 31
B. 31 and 32
C. 32 and 33
D. 33 and 34
E. 34 and 35
Solution:
Note that ∠ABC=90∘, so △ABC is a 45−45−90∘ triangle. Because hypotenuse AC=102​, the legs of △ABC have length 10. Therefore AB=10 and BD=BC+CD=10+20=30. By the Pythagorean Theorem,
AD=102+302​=1000​
Because 312=961 and 322=1024, it follows that (B)31<AD<32​.
