B Apply

19. Think About a Plan In the figure below, P is the incenter of isosceles Δ R S T . cap delta r s t .  What type of triangle is Δ R P T ? cap delta r p t question mark  Explain.

    

    • What segments determine the incenter of a triangle?
    • What do you know about the base angles of an isosceles triangle?

Constructions Draw a triangle that fits the given description. Then construct the inscribed circle and the circumscribed circle. Describe your method.

20. right triangle, Δ D E F cap delta d e f
21. obtuse triangle, Δ S T U cap delta s t u
22. Algebra In the diagram below, G is the incenter of Δ D E F , m ∠ D E F = 60 , cap delta d e f comma m angle d e f , equals 60 , comma  and m ∠ E F D = 2 · m ∠ E D F . m angle e f d , equals 2 , middle dot m angle e d f .  What are m ∠ D G E , m ∠ D G F , m angle d g e comma m angle d g f comma  and m ∠ E G F ? m angle e g f question mark

    

23. Writing Ivars found an old piece of paper inside an antique book.

    It read,

    From the spot I buried Olaf's treasure, equal sets of paces did I measure; each of three directions in a line, there to plant a seedling Norway pine. I could not return for failing health; now the hounds of Haiti guard my wealth. —Karl

    After searching Caribbean islands for five years, Ivars found an island with three tall Norway pines. How might Ivars find where Karl buried Olaf's treasure?

24. Proof Use the diagram below to prove the Concurrrency of Angle Bisectors Theorem.

    

    Given: Rays l, m, and n are bisectors of the angles of Δ A B C . cap delta eh b c .  X is the intersection of rays l and m, X D ¯ ⊥ A C ¯ , X E ¯ ⊥ A B ¯ , x d bar , up tack . eh c bar , comma . x e bar , up tack . eh b bar , comma  and X F ¯ ⊥ B C ¯ . x f bar , up tack , b c bar , .

    Prove: Ray n contains point X, and X D = X E = X F . x d equals . x e equals x f .

25. Noise Control You are trying to talk to a friend on the phone in a busy bus station. The buses are so loud that you can hardly hear. Referring to the figure below, should you stand at P or C to be as far as possible from all the buses? Explain.

    

Reasoning Determine whether each statement is true or false. If the statement is false, give a counterexample.

26. The incenter of a triangle is equidistant from all three vertices.
27. The incenter of a triangle always lies inside the triangle.
28. You can circumscribe a circle about any three points in a plane.
29. If point C is the circumcenter of Δ P Q R cap delta p q r  and the circumcenter of Δ P Q S , cap delta p q s comma  then R and S must be the same point.

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