问题1. ABC是一个三角形。在ΔABC内部找到一个与ΔABC的所有顶点等距的点。
解决方案：
To obtain a point which is equidistant from all vertices of a triangle we construct perpendicular bisectors of all sides (AB, BC, CA) of the triangle (ΔABC). The point of intersection of these bisectors is known as Circumcentre(O) which is equidistant from all vertices.
问题2.在三角形的内部找到一个与三角形所有边等距的点。
解决方案：
To obtain a point which is equidistant from all sides of a triangle we construct angle bisectors of all angles present in ΔABC i.e ∠BAC, ∠ABC, ∠ACB. The point of intersection of these bisectors is called Incentre(I) which is equidistant from all sides.
问题3.在一个巨大的公园中，人们集中在三个方面：
答：如果有不同的儿童滑梯和秋千，
B：人造湖位于附近，
C：靠近大型停车场和出口。
应该在哪里设置冰淇淋店，以使最多人数可以接近冰淇淋店?
(提示：客厅应与A，B和C等距)
解决方案：
The ice-cream parlour must be set somewhere so that it’s easily available for the public. So for such point it should be at a equal distance from point A, B, C & such point is termed as circumcentre.
问题4.通过尽可能多地填充边长为1 cm的等边三角形来完成六角形和星形Rangolies。计算每种情况下的三角形数量。哪个有更多三角形?
解决方案：
We need to find the number of triangles that can get fit the above figures i.e the hexagon and the star.
So,
Area of hexagon = (Area of small triangle inside hexagon) * 6
Area of small equilateral triangle = √3/4 * a2
= √3/4 * 52
= √3/4 * 25 = 25√3/4
So,
Area of hexagon = 25√3/4 * 6
= 150√3/4 cm2
Area of Star = Area of 6 triangles and 1 hexagon
= 6 * 25√3/4 + 150√3/4
= 300√3/4 cm2
Area of triangles of 1cm side that are to be fitted = √3/4 * 12
= √3/4 cm2
Number of triangles that can be accommodated inside hexagon and stars :
a. For Hexagon : Area of hexagon/ Area of 1cm side triangle
= 150√3/4 cm2 / √3/4 cm2
= 150 triangles
b. For Star : Area of star/ Area of 1cm triangle
= 300√3/4 cm2 / √3/4 cm2
= 300 triangles
Hence, the star can accommodate 150 more triangles than the hexagon.