Final answer:
Using the Pythagorean theorem, we find that cutting across the neighbor's yard would be a diagonal path of 25 feet, thus making the trip 10 feet shorter than the original route which totals 35 feet.
Step-by-step explanation:
To calculate how much shorter the student's walk would be if she could cut across the neighbor's yard on a diagonal, we can use the Pythagorean theorem. The student's initial path forms a right triangle, with the two sides of the triangle represented by the distances she walks: 20 feet to the corner of her street and then 15 feet to her friend's house. We can consider these two sides as the legs of the right triangle and the diagonal as the hypotenuse.
The Pythagorean theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c². Plugging in the values from the student's walk, we have:
20² + 15² = c²
400 + 225 = c²
625 = c²
To find the length of c, we take the square root of 625:
c = √625
c = 25 feet
Therefore, the diagonal path through the neighbor's yard would be 25 feet. To find how much shorter this path is compared to the student's usual route, we subtract the diagonal's length from the total distance walked on the usual route (20 feet + 15 feet):
35 feet - 25 feet = 10 feet shorter.