# In triangle CAT, a=4, c=5, and Cos T=1/8. What is the length of t?

malagala | College Teacher | (Level 2) Adjunct Educator

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In the triangle CAT, the angle T which is denoted by`/_T`

is adjacent to the sides a and

As the side c is opposite to the `/_C` ,we can draw a perpendicular to the side c from `/_C`

Then we can create a right angle trinagle with side a as the hypotenuse and the perpendicular line is denoted by t'` `

The other two side of this new right angle triangle are a and c'

Then we can write,

`cos T =(c')/(a) =(1)/(8)`

as a = 4, c' = 1/2

by using the pythogoras theorm,

`t'^(2) =a'^(2) -c'^(2)`

`t'^(2) = 4^(2) -(1)/(2)^(2)`

`t' = sqrt(63)/2`

` `

Then in the CAT triangle when we created a right angle triangel by drawing t' we have created another right angle triangle with side t as the hypotenuse

we can apply the pythogoras theorm to that right angle side as well

`t^(2) = t'^(2) + (c-c')^(2)`

`t^(2) = (63)/(4) + (5 -(1)/(2))^(2)`

`t^(2) = (63/4) + ( 9/2)^(2)`

`t^(2) = (63/4) + (81/4)`

`t^(2) = (144/4)`

t = 6