Exercise 16 Page 643

Let's start by finding the y-coordinates of the unlabeled vertices.

Both O and U are on the x-axis so their y-coordinate is 0. Also, S and T are 2k above the x-axis which must mean they have y-coordinates of 2k. Finally, since R bisects SU, we know that R must have a y-coordinate that is half of 2k.

Next, we will find the x-coordinates of our points. Since O is at the origin, it's x-coordinate must be 0. Also, since OU=UR=RS=ST, we know that the points along SU have x-coordinates of k and T has an x-coordinate of 2k.

When we know the coordinates of O and T, we can calculate the length of OT with the Distance Formula.

d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)

SubstitutePoints

Substitute ( 2k,2k) & ( 0,0)

d_(OT) = sqrt(( 2k - 0)^2 + ( 2k - 0)^2)

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Simplify right-hand side

SubTerms

Subtract terms

d_(OT) = sqrt((2k)^2 + (2k)^2)

PowProdII

(a b)^m=a^m b^m

d_(OT) = sqrt(4k^2 + 4k^2)

AddTerms

Add terms

d_(OT) = sqrt(8k^2)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

d_(OT) = ksqrt(8)