# Difference between revisions of "2012 AMC 10B Problems/Problem 12"

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− | If point B is due east of point A and point C is due north of point B, <math>\angle CBA</math> is a right angle. And if <math>\angle BAC = 45^\circ</math>, <math>\triangle CBA</math> is a 45-45-90 triangle. | + | If point B is due east of point A and point C is due north of point B, <math>\angle CBA</math> is a right angle. And if <math>\angle BAC = 45^\circ</math>, <math>\triangle CBA</math> is a 45-45-90 triangle. Thus, the lengths of sides <math>CB</math>, <math>BA</math>, and <math>AC</math> are in the ratio <math>1:1:\sqrt 2</math>, and <math>CB</math> is <math>10 \sqrt 2 \div \sqrt 2 = 10</math>. |

## Revision as of 19:42, 25 February 2012

## 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 , and . Point D is 20 meters due north of point C. The distance AD is between which two integers?

## Solution

If point B is due east of point A and point C is due north of point B, is a right angle. And if , is a 45-45-90 triangle. Thus, the lengths of sides , , and are in the ratio , and is .