Selected questions for the finale of elementary and junior high school mathematics, including real questions and detailed analysis of elementary and junior high school mathematics. It aims to broaden students' knowledge and expand their thinking. If you find this material helpful, you can share it with your classmates and friends. Thank you for your support and companionship, and I wish you progress in your studies!
On top of the right-angled isosceles triangle abc is a 1 4 circle part stacked on top of it, as shown below. It is known that ab=8cm, and the point d is the midpoint of ab, find the area of the shaded part.
Subtract the overlapping area from the area of 1 4 circles to get the shaded area.
The radius of 1 4 circle is the high dc on the hypotenuse of the right angle and other triangles abc, i.e. 8 2 = 4 (cm).
Its area is: 4 4 314×1/4=12.56 (square centimeters).
Then try to find the area of the overlapping part of the plot.
To do this, first find the area of the right angle equal triangle abc: 8 (8 2) 2 = 16 (square centimeters).
If the 1 4 circle is rotated clockwise around point D until its two radii are perpendicular to the two right-angled sides of the right-angled isosceles triangle ABC, as shown in the figure below:
Looking at the figure, the quadrilateral of the original overlap has been transformed into a square decf, but the size of its area has not changed.
At this point, if the DC is connected, as shown in the following figure:
It can be seen that the right angle equal to the triangle ABC has been divided into four identical small right angles equal to the triangle.
Therefore, the area of the overlapping part is: 16 2 4 = 8 (square centimeters).
So, the area of the shaded part of the figure is: 1256-8=4.56 (square centimeters).
List of high-quality authors