Answer:
CM=20 and CP=12
Step-by-step explanation:
The given triangle ΔACM has the measurements as follows:
m∠C=90°, CP⊥AM, AC=15, AP=9, PM=16.
To Find: CP and CM
We can use Pythagoras theorem to calculate the sides CP and CM.
Pythagoras theorem gives a relation between hypotenuse, base and height/perpendicular of a right angled triangle which is as follows:
[tex]h^{2}=p^{2}+b^{2}[/tex]
where h is hypotenuse of triangle, b is base and p is perpendicular of triangle.
The figure shows that in ΔACM is a right angled triangle at C where,
AM --> hypotenuse
CM --> base
AC --> height
So substituting values into formula:
[tex]AM^{2}=AC^{2}+CM^{2}[/tex]
[tex]25^{2}=15^{2}+CM^{2}[/tex]
[tex]625-225=CM^{2}[/tex]
[tex]400=CM^{2}[/tex]
[tex]\sqrt{400} =CM[/tex]
[tex]CM=20[/tex], which is required answer.
Similarly, we can see that triangle ΔCPM is also a right angled triangle at P and thus Pythagoras theorem can again be applied to calculate CP. Since CM is the side opposite to right angle P, it is the hypotenuse.
So we have,
[tex]CM^{2}=PM^{2}+CP^{2}[/tex]
[tex]20^{2}=16^{2}+CP^{2}[/tex]
[tex]400-256=CP^{2}[/tex]
[tex]144=CP^{2}[/tex]
[tex]\sqrt{144} =CP[/tex]
[tex]CP=12[/tex], which is required answer.
