Answer:
MT = 8.9 cm
Step-by-step explanation:
In the rectangle ABCD, Angle ABC = 90 degree
In the triangle ABC, Angle B = 90 degree
=> ABC is the right angle triangle
According to the Pythagoras theorem, we have the following equation:
+) [tex]AB^{2} + BC^{2} = AC^{2}[/tex]
⇔[tex]6^{2} + 7^{2} = AC^{2}[/tex]
⇔[tex]AC^{2} = 36 + 49 = 85[/tex]
⇔ [tex]AC = \sqrt{85}[/tex] cm
In the rectangle ABCD, M is the midpoint, so that it is also the midpoint of line segment AC.
=> AM = [tex]\frac{AC}{2}=\frac{\sqrt{85} }{2}[/tex] cm
TM is the height of the pyramid, so that it is perpendicular to the base ABCD.
As AM belongs to the surface of the rectangle ABCD
=> TM is also perpendicular to AM
=> AMT is the right-angled triangle with Angle AMT = 90 degree
According the Pythagoras theorem, we have the following equation:
[tex]AM^{2} + MT^{2} = AT^{2}[/tex]
⇔ [tex]MT^{2} = AT^{2} - AM^{2} = 10^{2} - (\frac{\sqrt{85} }{2} )^{2}[/tex]
⇔[tex]MT^{2} = 100 - \frac{85}{4} = \frac{315}{4}[/tex]
⇔[tex]MT = \sqrt{\frac{315}{4} } =\frac{3\sqrt{35} }{2}[/tex] ≈ 8.9 cm
MT = 8.9 cm
