Question

Let ABCD be a rectangle with AB = 4 and BC = 1. Denote by M the midpoint of line segment AD and by P the leg of the perpendicular from B onto CM.

a) Find the lengths of P B and PM.

b) Find the area of ABPM.

c) Consider now ABCD being a parallelogram. Denote by M the midpoint of side AD and by P the leg of the perpendicular from B onto CM. Prove that AP = AB. Hint: Extend line MC and denote by S its intersection with line AB and observe that two congruent triangles are formed.

Answer #1

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In the rectangle ABCD, AB = 6 and BC = 8. The diagonals AC and
BD intersect at O. Point P lies on the diagonal AC such that AP =
1. A line is drawn from B through P and meets AD at S. Let be R a
point on AD such that OR is parallel to BS. a) Find the lengths of
AS and RD. Hint: Denote AS = x. Use P S k OR and OR k BS...

Suppose M is the Midpoint of segment AB, P is the midpoint of
segment AM, and Q is the midpoint of segment PM. If a and b are the
coordinates of points A and B on a number line, find the
coordinates of P and Q in the terms of a and b.

Consider a cicle with AB as diameter and P another point on the
circle. Let M be the foot of the perpendicular from P to AB. Draw
the circles which have AM and M B respectively as diameters, which
meet AP at Q are BP are R. Prove that QR is tangent to both
circles.
Hint: As well as the line QR, draw in the line segments M Q and
M R.

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