Lesson 3 Cyclic Polygons Solidify Understanding

Ready

1.

The three angle bisectors of triangle $E F G$ are shown. Use a compass to construct an inscribed circle with center $H$ .

2.

The three angle bisectors of triangle $L M N$ are shown. Use a compass to construct an inscribed circle with center $O$ .

3.

The three angle bisectors of triangle $U V T$ are shown. Use a compass to construct an inscribed circle with center $S$ .

4.

Triangle $X Y Z$ has an inscribed circle with center $W$ . Identify the three segments that are tangent to circle $W$ .

5.

Triangle $A B C$ has an inscribed circle with center $D$ . Identify the three segments that are tangent to circle $D$ .

Set

Find the degree measure of the angle or the intercepted arc indicated in each figure.

6.

$⨀ M$ with $m ∠ L M N = 110 °$

$m \overset{⌢}{L N} =$
$m ∠ O L N =$
$m \overset{⌢}{O L} =$

7.

$⨀ B$ with $m ∠ A B C = 130 °$

$m \overset{⌢}{A C} =$
$m ∠ C A D =$
$m \overset{⌢}{D A} =$

8.

$⨀ F$

$m \overset{⌢}{E G} =$
$m \overset{⌢}{E H G} =$
$m ∠ G E H =$
$m ∠ G F H =$

9.

$⨀ M$ with diameter $\overset{―}{N K}$

$N L = 12$ and $K L = 5$

$N K =$
$m \overset{⌢}{N L K} =$
$m \overset{⌢}{N J K} =$
$m ∠ N L K =$
$m \overset{⌢}{K L} =$
$m \overset{⌢}{N L} =$

10.

How can a triangle be used to show the connection between an inscribed angle and the angle measure of the arc it intercepts? What is true about the angle measures in any triangle? What is true about the arc measure for an entire circle?

Go

Write the trigonometric equation needed to find $x$ and the trigonometric equation to find $y$ in each right triangle.

Then solve for $x$ ‌ and $y$ .

11.

12.

13.

14.

15.

The Pythagorean theorem could be used to check the side lengths in the previous problems. Examine the Pythagorean theorem for problems 13 and 14 shown.

13: ${10.7}^{2} + 9^{2} = 195.49, \sqrt{195.49} = 13.98$
14: ${89.4}^{2} + {67.4}^{2} = 12,535.12, \sqrt{12,535.12} = 111.96$

Explain why the answer in problem 13 doesn’t equal $14 meters$ and the answer in problem 14 doesn’t equal $112 feet$ .