Question

1. taking a shortcut look at the diagram shown. suppose you are walking south on the sidewalk of pine street. when you reach pleasant street, you cut across the empty lot to go to the corner of oak hill avenue and union street. explain why this route is shorter than staying on the sidewalks.

Explanation:

Step1: Apply the triangle inequality theorem

The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Step2: Analyze the two - path situation

Let the two - sidewalk path be the sum of two sides of a right - angled quadrilateral (which can be considered as two sides of a triangle - like structure in terms of path length comparison). The shortcut is the length of the hypotenuse (diagonal) of the right - angled quadrilateral.
If we assume the two sides of the "triangle - like" path (sidewalks) are \(a\) and \(b\), and the shortcut (diagonal) is \(c\). According to the triangle inequality theorem \(a + b>c\) (in a right - angled triangle \(a^{2}+b^{2}=c^{2}\), and for non - zero positive \(a\) and \(b\), \(a + b=\sqrt{a^{2}+ 2ab + b^{2}}>\sqrt{a^{2}+b^{2}}=c\)).

Answer:

This route is shorter than staying on the sidewalks because of the triangle inequality theorem. The sum of the lengths of the two sidewalk paths (the two sides of a right - angled "triangle - like" structure formed by the streets) is greater than the length of the shortcut (the hypotenuse of the right - angled "triangle - like" structure).