Area of triangle DUO: A=?
Formula of the area of a triangle: A=(1/2)bh
Base of the triangle: b
Height of the triangle: h
In this case:
b=OD=OM+MD=4+10→b=14
h=MU=?
Since DR is perpendicular to OP:
Angle RDO = 90° = Angle RDP
Like angle RDU is congruent with angle RDA, the angle UDM must be congruent with angle ADP.
If angle UMO is 90°, the angle UMD is 90° too.
The triangles ADP and UDM have two congruent angles:
Angle UMD = 90° = Angle APD
Angle UDM = Angle ADP
Then the triangles UDM and ADP are similars, and theirs sides must be proportionals:
MU/MD=AP/DP
Replacing the known values:
MU/10=4.5/3.75
MU/10=1.2
Solving for MU. Multiplying both sides of the equation by 10:
10(MU/10)=10(1.2)
MU=12
A=(1/2)OD*MU
A=(1/2)(14)(12)
A=84
Answer: The area of triangle DUO is 84.00 square units.