Grades 7-8 Video Solutions 2024-2024_7-8

Video Transcription

Problem number 23. The diagram shows three semicircles inside a rectangle. The middle semicircle touches the other two semicircles, which in turn each touch a shorter side of the  rectangle. The largest semicircle also touches one of the longer sides of the rectangle. The  shortest distances from that side of the rectangle to the other two semicircles are 5 centimeters  and 7 centimeters respectively, as shown. What is the perimeter in centimeters of the rectangle?  So let's make this diagram a bit larger and add a dimension h centimeters for the height of this  rectangle. And notice that we can move this height along the rectangle and get some additional  heights h, h minus 5, and h minus 7. And these come from the fact that this height is the same  throughout the rectangle. But we also notice that this height is the radius of the circle or the  semicircle, because that is the distance from the center of the semicircle to the topmost point,  which is at the middle of the semicircle. That means that these semicircles have radii h, h minus  5, and h minus 7, as shown in the diagram. And so now we should use the distance of 36 centimeters  to our advantage. Let's find the total distance in terms of the radii of the semicircles.  The first radii will give 2h, because that is the diameter of the first semicircle. The next one  will give 2 times quantity h minus 5, and the last one will give 2 times quantity h minus 7.  And we know that all together, these three should add up exactly to 36. And so if we factor out a 2,  we get 36 equals 2 times the quantity h plus h minus 5 plus h minus 7. Dividing by 2 gives 18  equals h plus h minus 5 plus h minus 7. And collecting terms, we get 18 is equal to 3h minus 12.  That means that 30 is equal to 3h, or h is equal to 10. But we want the perimeter of the rectangle,  and so we know that the length is 36 and the height is 10. So the perimeter is 2 times 10 plus 36, and that is equal to 92. So the answer is B, 92.

Video Summary

The problem involves finding the perimeter of a rectangle containing three semicircles. The semicircles have radii defined by the rectangle's height \( h \), and distances between the semicircles are given. Using given dimensions, equations are formed to calculate the radii as 10 cm. The conclusion leads to calculating the rectangle's perimeter based on its deduced height (10 cm) and provided length (36 cm). The final perimeter is calculated as \( 2 \times (10 + 36) = 92 \) cm. Thus, the rectangle's perimeter is 92 cm.