it is always possible to draw a circle which passes through all four vertices of a rectangle. Explain why.
The center of the rectangle is equidistant from each vertex. If you make the center of your circle the center of the rectangle, then the four vertices of the rectangle will be equidistant from the center of the circle as well, making each distance from the vertex of the rectangle to the center a radius of the circle.
Well the answer is quite simple.
The vertices of the rectangle is the same as the corner of each circle and it also has an equal distance. The center of the circle and the center of the rectangle is the same then the distance from the center to the vertex of the rectangle is the radius......
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The center of the rectangle is equidistant from each vertex. If you make the center of your circle the center of the rectangle, then the four vertices of the rectangle will be equidistant from the center of the circle as well, making each distance from the vertex of the rectangle to the center a radius of the circle.
if m<6=72,then m<7=?
Well, it's actually not always possible to draw a circle that passes through all four vertices of a rectangle. That would be quite a remarkable feat, don't you think? It would be like trying to fit a square peg into a round hole but on a larger scale.
But hey, don't be disappointed! Just because this circle-dream may not always come true for rectangles, it doesn't mean circles can't have fun with them. After all, circles and rectangles can still be good friends - they just might not be able to hug each other perfectly every time!
To understand why it is always possible to draw a circle that passes through all four vertices of a rectangle, let's break it down step by step:
1. Start with a rectangle: A rectangle is a four-sided polygon with opposite sides being equal in length. Its angles are always right angles (90 degrees).
2. Circumscribe the rectangle: To draw a circle passing through all four vertices of a rectangle, imagine a circle that is big enough to enclose the entire rectangle. This circle is called a circumscribed or circumcircle.
3. Center of the circle: The center of the circle lies at the intersection of the diagonals of the rectangle. The diagonals are the line segments that connect opposite vertices of the rectangle, dividing it into four equal triangles. Since the diagonals bisect each other, their point of intersection is the center of the circle.
4. Radius of the circle: To find the radius of the circumcircle, we need to measure the distance from the center to any of the four vertices. Since the diagonals bisect each other, they also divide the rectangle into two congruent triangles. Therefore, the radius of the circumcircle is equal to half the length of the diagonal.
5. The perpendicular bisectors: To visualize the circle better, construct the perpendicular bisectors of the rectangle's sides. A perpendicular bisector is a line that divides a line segment into two equal halves while being perpendicular to that line segment. When drawn for all four sides of the rectangle, these bisectors will intersect at the center of the circumscribing circle.
6. Verification: By drawing the circle using the obtained center and radius, it will pass through all four vertices of the rectangle. This can be verified by checking that the distance between the center of the circle and each vertex is equal to the radius.
In summary, it is always possible to draw a circle passing through all four vertices of a rectangle by finding the center of the circumcircle at the intersection of the diagonals and determining the radius as half the length of any diagonal. The circle can be further verified by checking that it passes through all four vertices.