L.8 Perimeter of polygons with an inscribed circle

Perimeter of polygons with an inscribed circle

Key Notes :

A tangent to a circle is a line that intersects a circle at exactly one point. The point where the line intersects the circle is called the point of tangency.

A circle is shown. There is also a line that intersects the circle at exactly one point. The line is labeled tangent line, and the point where the line intersects the circle is labeled point of tangency.

If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.

For example, BC is tangent to ⨀A at point B. So, BC is perpendicular to radius AB at point B.

Circle A is shown. Line BC intersects circle A at one point, point B. Radius AB is perpendicular to line BC at point B.

The converse is also true. If a line is perpendicular to the radius of a circle at a point on the circle, then the line is tangent to the circle.

Solving problems involving tangents

You can use what you know about tangents to solve problems.

Determining if a line is tangent to a circle

Let’s try it! Determine if KL is tangent to ⨀M.

Circle M is shown. Triangle KLM is also shown. One of the sides of the triangle, line segment KM, is a radius of circle M. Point L is outside of circle M. Line segment KM is labeled 15 centimeters, line segment KL is labeled 8 centimeters, and line segment LM is labeled 17 centimeters.

If KL is perpendicular to the radius KM, then KL is tangent to ⨀M. So, check whether ∠K is a right angle. The converse of the Pythagorean theorem states that △KLM is a right triangle with hypotenuse LM if KL²+KM²=LM².

KL²+KM² ≟ LM²

8²+15²≟17²

64+225≟289 Square.

289=289 Add.

Since KL²+KM²=LM² . △KLM is a right triangle with hypotenuse LM . Therefore, ∠K is a right angle, and KL is perpendicular to KM is tangent to ⨀M.

Finding the radius of a circle

ST is tangent to ⨀R. Find the radius of the circle.

Circle R is shown. Right triangle RST is also shown. Triangle RST has leg RT, which is a radius of circle R. Leg RT is labeled x. Point S is outside of circle R. The part of line segment RS that is a radius of circle R is labeled x, and the part of line segment RS that lies outside of the circle is labeled 9 inches. Line segment ST is labeled 21 inches.

Since ST is tangent to ⨀R, ST is perpendicular to RT. So, △RST is a right triangle with hypotenuse RS. You can use the Pythagorean theorem to find the radius of the circle, x.

RT²+ST²=RS²

x²+21²=(x+9)² Plug in RT=x, ST=21, and RS=x+9.

x²+441=x²+18x+81 Square 21 and (x+9).

441=18x+81 Subtract x² from both sides.

360=18x Subtract 81 from both sides.

20=x Divide both sides by 18.

So, the radius of ⨀R is 20 inches.

Circles with two tangents

If two line segments are tangent to a circle and share a common endpoint outside the circle, then the line segments are congruent.

For example, XW and XY are both tangent to ⨀Z. Those line segments have a common endpoint, X, which lies outside of ⨀Z. So, XW≅XY.

Solving problems with two tangents

Let’s try it! DE and EF are tangent to ⨀G. Find DE.

DE and EF are tangent to ⨀G. Also, DE and EF have a common endpoint, E, that lies outside of circle ⨀G. So, DE≅EF. Set DE equal to EF and solve for x.

DE=EF

2x+3=3x–1 Plug in DE=2x+3 and EF=3x–1

2x+4=3x Add 1 to both sides.

4=x Subtract 2x from both sides.

Now, find DE. Plug in x=4 and solve for DE.

DE=2x+3

=2(4)+3 Plug in x=4.

=11 Simplify.

So, DE=11.

Constructing a tangent to a circle

You can construct a tangent to a circle using a straightedge and a compass.

Tangent that passes through a given point on the circle

To construct the tangent to a circle that passes through a given point on the circle, start with the circle, center, and point on the circle. Call the center point A and the point on the circle point B.

Circle A is shown. Point B is on circle A.

Now, use your straightedge to draw a line segment with endpoint A that passes through point B. Extend the line segment outside of the circle.

The diagram from above is shown with an additional line segment. The line segment has endpoint A, passes through point B, and extends outside of circle A.

Then, construct a line segment that has point B as a midpoint. Set your compass to a setting shorter than AB and place the point of your compass at point B. Draw two arcs using that setting. One of the arcs should intersect the line segment inside of the circle, and the other arc should intersect the line segment outside of the circle. Mark the points where the arcs intersect the line segment. Call them points C and D. You now have CD, which has midpoint B.

The diagram from above is shown with two additional arcs that intersect line segment AB. The arcs were created using a compass with its point placed on point B and a setting shorter than line segment AB. One of the arcs is inside of circle A and intersects line segment AB at point C. The other arc is outside of circle A and intersects line segment AB at point D.

The remainder of the steps involve constructing the perpendicular bisector of CD. Set your compass to a setting greater than 1/2CD and place the point of your compass at point C. Draw an arc near CD.

The diagram from above is shown with an additional arc. That additional arc is above line segment AB and it was created using a compass with its point placed on point C and a setting greater than half the length of line segment CD.

Next, using the same compass setting, place the point of your compass at point D. Draw an arc that intersects the arc you drew in the previous step. Mark the point where the arcs intersect. Call it point E.

The diagram from above is shown with an additional arc and an additional point. The additional arc intersects the arc above line segment AB and was created using the same compass setting as before with the compass point placed on point D. The point where the arcs above line segment AB intersect is labeled E.

Last, use your straightedge to draw a line that connects points B and E.

The diagram from above is shown with an additional line. The line passes through point B and point E.

BE is the perpendicular bisector of CD, so it is perpendicular to radius AB. Thus, BE is tangent to ⨀A at point B.

Tangent that passes through a given point outside of the circle

You can also construct a tangent to a circle that passes through a given point outside of the circle. Start with the circle, center, and point outside of the circle. Call the center point P and the point outside of the circle point Q.

Circle P is shown. Point Q is outside of circle P.

Now, use your straightedge to draw a line segment with endpoints P and Q.

The diagram from above is shown with an additional line segment. The line segment has endpoints P and Q.

The next few steps involve constructing the perpendicular bisector of PQ. Set your compass to a setting greater than 1/2PQ and place the point of your compass at point P. Draw arcs above and below the line segment.

The diagram from above is shown with two additional arcs. The arcs are above and below line segment PQ and were created using a compass with its point placed on point P and a setting greater than half the length of segment PQ.

Next, using the same compass setting, place the point of your compass at point Q. Draw arcs above and below the line segment. Mark the points where the arcs intersect. Call them points R and S.

The diagram from above is shown with two additional arcs and two additional points. Those additional arcs are above and below line segment PQ and were created using a compass with its point placed on point Q using the same compass setting as before. The arcs above line segment PQ intersect at point R. The arcs below line segment PQ intersect at point S.

Then, draw a line that connects points R and S using your straightedge. This is the perpendicular bisector of PQ. Mark the point where RS intersects PQ. Call it point T. RS is the perpendicular bisector of PQ, so point T is the midpoint of PQ.

The diagram from above is shown with an additional line. The line that passes through point R and point S. Line RS intersects line segment PQ at point T.

After that, construct the point where the tangent intersects the circle. Set your compass to a setting equal to PT and place the point of your compass at point T. Draw an arc that intersects the circle. Mark the point where the arc intersects ⨀P, and call it point U.

The diagram from above is shown with one additional arc and one additional point. The additional arc intersects circle P at point U. The arc was created using a compass with its point placed on point T using a setting equal to the length of line segment PT.

Last, use your straightedge to draw a line segment that has endpoints Q and U.

The diagram from above is shown with one additional line segment. The additional line segment is line segment QU, which is tangent to circle P at point U.

QU is tangent to ⨀P at point U.

Why does this construction work?

To understand why QU is tangent to ⨀P, you can draw the entire circle for the arc that intersected ⨀P. This circle is centered at point T. Since ⨀T was created with a radius of PT and point T is the midpoint of PQ, PQ is a diameter of ⨀T.

The diagram from above is shown except the arc that intersects circle P at point U has been extended to create a circle centered at point T. Points P and Q lie on circle T.

You can also add PU to create ∠PUQ. Since ∠PUQ is inscribed in a semicircle, it must be a right angle.

The diagram from above is shown except with one additional line segment, line segment PU. Line segment PU is a radius of circle P. Angle PUQ is a right angle.

So, QU is perpendicular to PU at point U. Notice that PU is a radius of ⨀P. Since QU is perpendicular to a radius of ⨀P at a point on the circle, QU is tangent to ⨀P.

Learn with an example

What is FG?