ABCA1B1C1 is a regular triangular prism, all edges of which are 8 cm. Points K and P are the midpoints of edges A1C1

| August 16, 2021 | education

ABCA1B1C1 is a regular triangular prism, all edges of which are 8 cm. Points K and P are the midpoints of edges A1C1 and C1B1, respectively. Calculate the perimeter of the quadrilateral AKPB.

Since the points K and P are the middle of the segment A1C1 and B1C1, then the segment KP is the midline of the triangle A1B1C1, and therefore its length is equal to half the length of the side A1B1.

KP = A1B1 / 2 = 8/2 = 4 cm.

In a right-angled triangle AA1K, the hypotenuse of AK will be equal to:

AK ^ 2 = AA1 ^ 2 + A1K ^ 2 = 8 ^ 2 + 4 ^ 2 = 64 + 16 = 80.

AK = 4 * √5 cm.

Similarly, from the triangle BB1R BP = 4 * √5 cm.

Then the perimeter of the quadrangle AKPB = AB + KP + AK + BP = 8 + 4 + 4 * √5 + 4 * √5 = 16 + 8 * √5 = 8 * (2 + √5) cm.

Answer: The perimeter of the quadrilateral is 8 * (2 + √5) cm.



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