Straight lines a and b are drawn through point M between the parallel planes alpha and beta. Line a intersects

| May 14, 2021 | education

Straight lines a and b are drawn through point M between the parallel planes alpha and beta. Line a intersects the alpha and beta plane at points A and A1, and line b at B and B1. Find the length ab if A1B1 = 18 cm and MB1 refers to MB1 as 2: 3.

Consider triangles ABM and A1B1M and prove that they are similar.

Angle АМВ and angle А1МВ1 are equal as vertical angles at the intersection of straight lines АА1 and ВВ1.

Since the alpha and beta planes are parallel, and the straight lines AB and A1B1 belong to these planes, AB is parallel to A1B1, and the angle BB1A1 is equal to the angle ABB1 as criss-crossing angles at the intersection of parallel AB and A1B1 secant BB1. Then triangles ABM and A1B1M are similar in two angles.

Let the length of the segment MВ1 = 2 * X cm, then MВ = 3 * X cm.

In similar triangles ABM and A1B1M1:

AB / A1B1 = MB / MB1.

AB = A1B1 * MВ / MB1 = 18 * 3 * X / 2 * X = 27 cm.

Answer: The length of the segment AB is 27 cm.



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