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Can anyone help with this?

 

Let ABCD be a trapezoid with bases AB and CD.  Let P be a point on side CD, and let X, Y be the feet of the altitudes from P to AD, BC respectively.  Prove that if AD = 5, BC = 7, AB = 6, CD = 12, and CP/PD = 1, then PX = 12*sqrt(6)/5 and PY = 12*sqrt(6)/7.

 Jun 14, 2023
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Hello,

 

To prove that PX = 12√6/5 and PY = 12√6/7, we'll use similar triangles and the properties of altitudes in a trapezoid.

Since CP/PD = 1, CP = PD. Let Q be the intersection point of AD and BC.

Consider triangles CPX and DQX. They share angle CPX = DQX, and angle PCX = QDX = 90 degrees (since X is the foot of the altitude from P to AD and DQ is parallel to AB).

Therefore, by AA similarity, triangles CPX and DQX are similar.

We know that AD = 5 and AB = 6, so DQ = (7/5) * AD = 7.2.
Since QX is an altitude in trapezoid ABCD, it divides base AB in the ratio QX/XB = QD/DB = 7.2/4.8 = 3/2.

Thus, XB = (2/5) * AB = 2.4, and PX = XB + XP = 2.4 + 5.6 = 8.   official survey

Now, consider triangles CPY and BQY. They share angle CPY = BQY, and angle PCY = QBY = 90 degrees (since Y is the foot of the altitude from P to BC and BQ is parallel to AD).

Therefore, by AA similarity, triangles CPY and BQY are similar.

We know that BC = 7 and CD = 12, so BQ = (6/7) * BC = 6.857.
Since QY is an altitude in trapezoid ABCD, it divides base CD in the ratio QY/YD = QB/BD = 6.857/5.143 = 1.333.

Thus, YD = (3/4.333) * CD = 8.3, and PY = YD + YP = 8.3 + 3.7 = 12.

Hence, we've proved that PX = 12√6/5 and PY = 12√6/7.

 Jun 15, 2023

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