Chapter 10.5, Problem 33PPS

To determine

Toprove: that the line is perpendicular to the radius of a circle at its endpoint and line is tangent to the circle.

Answer to Problem 33PPS

  $l$ must be tangent to circle $S$ .

Explanation of Solution

Given:

  $l$ is perpendicular to circle $s$ and $T;\overline{ST}$ is a radius of circle $s$ .

Concept used:

Side, side, side $\left(SSS\right)$:

Two triangles having same all three sides equal.

Side, angle, side $\left(SAS\right)$:

Two triangles having two sides equal and one angle equal.

angle, side, angle $\left(ASA\right)$:

Two triangles where two angles and one side is equal.

Calculation:

Assume that $l$ in not the tangent to circle $S$ , then it must intersect circle at a point other than $T$ .

Let the point be $U$ . Then the triangle $TSU$ must be an isosceles triangle since, $SU=ST$ .

  $\angle STU\text{ and }\angle TSU$ must be both acute because each has to be less than ${90}^0$ in order for $\angle TSU$ to have a positive degree measure. However, if $\angle STU$ is acute, the line $UT$ will not be perpendicular to radius $ST$ , which is contradictory to the given.

The diagram of the assumption is:

The assumption is false that it is not true that $l$ in not the tangent to circle $S$ .

Hence, $l$ must be tangent to circle $S$ .