Circumscribed and also inscribed circles are sketched roughly the circumcenter and the incenter

In this lesson fine look in ~ circumscribed and inscribed circles and the special relationships that kind from this geometric ideas.

You are watching: Many circles can be circumscribed about a given triangle


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Circumscribed circles

When a one circumscribes a triangle, the triangle is within the circle and also the triangle touch the circle v each vertex.


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You use the perpendicular bisectors of each side the the triangle to discover the the center of the circle that will circumscribe the triangle. So for example, given ??? riangle GHI???,


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The center allude of the circumscribed one is called the “circumcenter.”

For an acute triangle, the circumcenter is inside the triangle.

For a right triangle, the circumcenter is top top the side opposite best angle.

For one obtuse triangle, the circumcenter is exterior the triangle.

Inscribed circles

When a one inscribes a triangle, the triangle is exterior of the circle and also the circle touch the political parties of the triangle in ~ one allude on each side. The sides of the triangle room tangent come the circle.


To illustration an inscriptions circle within an isosceles triangle, usage the angle bisectors of each side to discover the facility of the circle that’s inscribed in the triangle. For example, offered ??? riangle PQR???,


Remember that each next of the triangle is tangent come the circle, for this reason if you draw a radius from the center of the circle to the allude where the circle touches the sheet of the triangle, the radius will form a best angle through the leaf of the triangle.

The center allude of the inscribed one is called the “incenter.” The incenter will always be within the triangle.

Let’s use what we know around these build to settle a couple of problems.


Finding the radius that the circle that circumscribes a trianle

Example

???overlineGP???, ???overlineEP???, and also ???overlineFP??? are the perpendicular bisectors the ???vartriangle ABC???, and also ???AC=24??? units. What is the measure of the radius the the circle the circumscribes ??? riangle ABC????


Point ???P??? is the circumcenter the the circle that circumscribes ??? riangle ABC??? due to the fact that it’s whereby the perpendicular bisectors that the triangle intersect. We can attract ???igcirc P???.


We likewise know the ???AC=24??? units, and since ???overlineEP??? is a perpendicular bisector the ???overlineAC???, allude ???E??? is the midpoint. Therefore,

???EC=frac12AC=frac12(24)=12???


Now we can draw the radius from allude ???P???, the center of the circle, to allude ???C???, a point on its circumference.


We have the right to use best ??? riangle PEC??? and the Pythagorean organize to resolve for the length of radius ???overlinePC???.

???5^2+12^2=(PC)^2???

???PC=13???


You usage the perpendicular bisectors of every side that the triangle to uncover the the facility of the circle that will certainly circumscribe the triangle.


Example

If ???CQ=2x-7??? and also ???CR=x+5???, what is the measure of ???CS???, provided that ???overlineXC???, ???overlineYC???, and ???overlineZC??? space angle bisectors of ??? riangle XYZ???.


Because ???overlineXC???, ???overlineYC???, and also ???overlineZC??? are angle bisectors the ??? riangle XYZ???, ???C??? is the incenter that the triangle. The circle with facility ???C??? will certainly be tangent to each side the the triangle in ~ the allude of intersection.

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???overlineCQ???, ???overlineCR???, and also ???overlineCS??? are all radii of circle ???C???, so they’re all equal in length.

???CQ=CR=CS???

We need to find the size of a radius. We recognize ???CQ=2x-7??? and ???CR=x+5???, so

???CQ=CR???

???2x-7=x+5???

???x=12???

Therefore,

???CQ=CR=CS=x+5=12+5=17???


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