Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

A square of size 30cm is divided into 9 identical smaller squares. The large square contains 3 circles with a radii 5cm on the bottom right, 4cm on the  top left, and 3cm on the top right, as shown. What is the area of this shaded part?  We notice something characteristic about the numbers. So any time that you're given numbers that might be Pythagorean triples, it should be  something that sets off bells, and here we note that the sum of the areas of the shaded circles is 4 squared pi plus 3 squared pi, which equals 25 pi.  Conveniently, the unshaded circle has area 5 squared pi, which is 25 pi. Therefore, the area of the shaded circles is the exact same as the area of the unshaded  circle, and then the area of the two shaded circles will fit exactly inside the unshaded circle's area. We can draw it like this.  As you can see, the two shaded circles are moved into the unshaded circle, and then we get a very very nice shape.  Because we were told that the side length is 30, and 5 out of 9 squares are shaded, then we can find the total shaded area as 5 ninths times 30 squared, which we know is  500, and our correct answer is therefore B.

Video Summary

A 30cm square is divided into 9 smaller squares and contains three circles with radii of 5cm, 4cm, and 3cm. The problem leverages Pythagorean triples and notes that the sum area of the two smaller circles (with radii 4cm and 3cm) matches the area of the largest circle (5cm). The combined area of these two smaller circles equals the large circle's area, demonstrating a neat fit. Given the square's side length is 30, with 5 out of 9 squares shaded, the total shaded area is calculated as \( \frac{5}{9} \times 30^2 = 500 \). The correct answer is option B.

Keywords

geometry

Pythagorean triples

circle area

square division

shaded area