When the two triangles are similar, the distance is minimized. Consider reflecting one of the vertical legs in the line that describes the top counter.
Diagram:
Is the perimeter below 26 feet? Yes!
Practice makes perfect
From the diagram, we see two right triangles where the horizontal legs have a combined length of 11 inches. If we call the length of one of these legs x, the other leg will have a length of 11-x.
To minimize the distance FK+KS, We can reflect S in AB and thereby creating △ KBS' which is congruent to △ KBS. We know this is the case by the SAS≅ condition.
The minimum distance between two points is a straight line. Therefore, we can minimize FK+KS' if K is placed along the top counter such that FS' is a straight line.
Since both FS' and AB are straight lines, △ FAK and △ S'KB will have a pair of vertical angles at K. Therefore, we know by the Vertical Angles Theorem that these angles are congruent.
This implies that the combined minimal distance from the stove and refrigerator is obtained when △ FAK and △ SBK are similar triangles. Since the ratio between corresponding sides is equal no matter what sides you choose, we can write the following equation.
11-x/x=6/3
Let's solve this equation for x.
If x= 3.67, the distance 11-x should be 11- 3.67=7.33 feet. Let's add this information to the diagram. Note that from problem 7-84, we know that FS is about 11.44 feet.
Now we have enough information to determine y and z with the Pythagorean Theorem.
7.33^2+6^2=y^2 ⇔ y≈ 9.47, y≥ 0
3.67^2+3^2=z^2 ⇔ z≈ 4.74, z≥ 0
Now we can find the combined length of the three distances.
Perimeter=4.74+9.47+11.44≈ 25.65 in.
Her design now conforms to the recommendations by the Kitchen & Bath Association.
