The mid points of an equilateral triangle are joined. Forming a second triangle, a third triangle is formed joining the midpoints of the second and the process continued infinitely. If the area of the first triangle is 24 units, then find the sum of areas of all triangles.
The mid points of an equilateral triangle are joined. Forming a second triangle, a third triangle is formed joining the midpoints of the second and the process continued infinitely. If the area of the first triangle is 24 units, then find the sum of areas of all triangles.
The correct option is C 32
If you join the mid-points of a triangle, you form 4 smaller equilateral triangles as follows
Thus, the shaded triangle (second triangle in the question) has an area exactly 14 of the area of the outer triangle. This process will continue and the next triangle which will be formed from the midpoints of the second triangle will have an area =14 of the second triangle
Thus this forms an infinite Geometric progression where the common ratio is 14 and the initial term (Area of first triangle)=24
S=a1−r=241−14=24×43=32 units. Answer is option (c)
32
If you join the mid-points of a triangle, you form 4 smaller equilateral triangles as follows
Thus, the shaded triangle (second triangle in the question) has an area exactly 14 of the area of the outer triangle. This process will continue and the next triangle which will be formed from the midpoints of the second triangle will have an area =14 of the second triangle
Thus this forms an infinite Geometric progression where the common ratio is 14 and the initial term (Area of first triangle)=24
S=a1−r=241−14=24×43=32 units. Answer is option (c)
