Question

Consider triangle TUV in the figure below.

The perpendicular bisectors of its sides are line AD , line BD and line CD . They meet at a single point D .
(In other words, D is the circumcenter of triangle TUV.)
Suppose CD = 30 , BV = 74, and UD = 78.
Find UV , VD , and TC.

Answer 1

BV = √5106, VD = √8336, and TC = √6256.

Since D is the circumcenter of △ TUV, it is equidistant to all three vertices of the triangle. Therefore, we can use the Pythagorean Theorem to solve for the missing side lengths.

To find BV, we can use the Pythagorean Theorem on right triangle BVD, where BV is the hypotenuse, DV is the opposite side, and BD is the adjacent side. We know that BD = 1/2 UV = 1/2 (142) = 71 and DV = CD/2 = 30/2 = 15. Plugging these values into the Pythagorean Theorem, we get:

BV^2 = DV^2 + BD^2

BV^2 = 15^2 + 71^2

BV^2 = 5106

BV = √5106

To find VD, we can use the Pythagorean Theorem on right triangle DVD, where VD is the hypotenuse, UV is the adjacent side, and UD is the opposite side. We know that UD = 78 and UV = 142. Plugging these values into the Pythagorean Theorem, we get:

VD^2 = UV^2 - UD^2

VD^2 = 142^2 - 78^2

VD^2 = 8336

VD = √8336

To find TC, we can use the Pythagorean Theorem on right triangle TDC, where TC is the hypotenuse, CD is the adjacent side, and UD is the opposite side. We know that UD = 78 and CD = 30. Plugging these values into the Pythagorean Theorem, we get:

TC^2 = CD^2 + UD^2

TC^2 = 30^2 + 78^2

TC^2 = 6256

TC = √6256

Therefore, BV = √5106, VD = √8336, and TC = √6256.