Question

The radius of circle N measures 25 cm and GH = 48 cm. What is TS?

14
24
9
18

Answer 1

18

Explanation:

The hint is suggesting you use the Pythagorean theorem. That will tell you ...

NH² = NT² +TH²

The segment NT is the perpendicular bisector of GH, so ...

GT = TH = 48/2 = 24.

If you like, you can also use ...

NT = NS -TS = NH -TS

So the above Pythagorean theorem equation can be written as ...

NH² = (NH -TS)² +TH²

NH² = NH² -2·NH·TS +TS² +TH²

0 = -2·25·TS +TS² +24²

(TS -18)(TS -32) = 0 . . . . . . factor the above equation

The values of TS that make the factors zero are TS = 18 and TS = 32.

We are interested in the solution for TS < 25, so ...

TS = 18

_____

The given side lengths of triangle NTH are 24 and 25. If you're familiar with Pythagorean triples, you know that NT must be 7, hence TS is 25-7 = 18. Even if you're not, you can find NT from the Pythagorean theorem:

NT = √(25² -24²) = √49 = 7

Answer 2

TS = 18

Explanation:

NR is the perpendicular bisector of GH, thus

TH = 48 ÷ 2 = 24

Using Pythagoras' identity in right triangle NTH

The square on the hypotenuse is equal to the sum of the squares on the other 2 sides, that is

NT² + 24² = 25², that is

NT² + 576 = 625 ( subtract 576 from both sides )

NT² = 49 ( take the square root of both sides )

NT =
`√(49)` = 7

Note that NS is the radius of the circle, thus

TS = NS - NT = 25 - 7 = 18