Let ABCD be a trapezoid with bases AB and CD. Let AD=5 and BC=7, and let P be a point on side CD such that CP/DP=7/5. Let X,Y be the feet of the altitudes from P to AD, BC respectively. Show that PX=PY.
To show that PX = PY, we can use the similarity of triangles and the concept of altitude in a triangle.
First, let's draw the trapezoid ABCD with the given information:
```
A ______ B
| |
| |
| |
| |
D ______ C
```
Given:
AD = 5
BC = 7
CP/DP = 7/5
Let's label the foot of the altitude of P on AD as X and the foot of the altitude of P on BC as Y:
```
A ______ B
| |
| __P
| / |
| / |
| / |
| / |
| X Y
D ______ C
```
To solve this problem, we will use the concept that if two triangles are similar, their corresponding sides are proportional.
We want to prove that PX = PY. To do this, we will show that triangles CPX and DPY are similar, which will imply that PX/PY = CP/DP = 7/5.
Consider triangles CPX and DPY.
We know that angle CPX = angle DPY = 90 degrees because PX and PY are altitudes in triangles CPX and DPY, respectively.
We also know that angle PXC and angle PYD are congruent angles. This can be proven because angle CDP and angle PXC are corresponding angles when line CP is parallel to line XY. The same applies to angle CPX and angle DPY.
Thus, we have:
- angle CPX = angle DPY
- angle PXC = angle PYD
- angle CPX = angle DPY because they are both 90 degrees
By the Angle-Angle (AA) similarity postulate, we can conclude that triangles CPX and DPY are similar.
Since triangles CPX and DPY are similar, their corresponding sides are proportional. In particular, we have PX/PY = CP/DP = 7/5. Thus, we have proven that PX = PY.
Therefore, PX = PY is shown.
Mr Frank has 1.03 kilograms of fertilizer for the plants in his nursery. He wants every plant to get 95 mg of fertilizer.
What is the number of plants he could fertilize with that amount?