# How do I show that the area of the triangle OAB is 36 1/8 ?

##### 3 Answers

#### Explanation:

#"area of "triangleOAB=1/2xx"base"xx"height"#

#color(white)(xxxxxxxxxxx)=1/2xxOBxxOA#

#"we require to find the coordinates of A and B"#

#"obtain the equation of the tangent line AB"#

#"remembering that the tangent is at right angles to"#

#"the radius"#

#"the slope of the radius between" (0,0)" and "(1,4)" is"#

#m_("radius")=(Deltay)/(Deltax)=4/1=4#

#"hence the slope of the tangent line is"#

#m_("tangent")=-1/m=-1/4#

#rArry=-1/4x+blarrcolor(blue)"is the partial equation"#

#"to find b substitute "(1,4)" into the partial equation"#

#4=-1/4+brArrb=4+1/4=17/4#

#rArry=-1/4x+17/4larrcolor(red)"equation of tangent line"#

#x=0rArry=17/4rArrA=(0,17/4)#

#y=0rArr-1/4x+17/4=0rArrx=17rArrB=(17,0)#

#"area of "triangleOAB=1/2xx17xx17/4#

#color(white)(xxxxxxxxxxxx)=289/8=36 1/8" units"^2#

Please refer to the **Explanation.**

#### Explanation:

Name the **point of contact**

Let, **radius** of the circle in question, say

Since, **centre** of

Clearly, the **eqn.** of

Recall that, the eqn. of tgt.

Hence,

This eqn. of

See the explanation below.

#### Explanation:

Let point C(1, 4) be the point of tangency then the radius OC:

The OC is also perpendicular to AB, draw two lines from C perpendicular to OB and OA at points D and and E respectively:

We can calculate the following angles:

OAD = arctan(1/4)=14 degrees

OCE = 90 - 14 = 76 degrees

Now consider two right triangles AEC and DCB:

Angle EAC and DCB are equal also angles ECA and DBC are equal, we can calculate the following angles:

EAC = DCB = 90 - 14 = 76 degrees

ECB = DBC = 90 - 76 = 14 degrees

We can calculate the following lengths:

then:

Finally the area of OAB is: