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EXAMPLE 3

EXAMPLE 3. Find the orthocenter. Find the orthocenter P in an acute, a right, and an obtuse triangle. SOLUTION. Right triangle P is on triangle. Acute triangle P is inside triangle. Obtuse triangle P is outside triangle.

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EXAMPLE 3

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  1. EXAMPLE 3 Find the orthocenter Find the orthocenter Pin an acute, a right, and an obtuse triangle. SOLUTION Right triangle Pis on triangle. Acute triangle Pis inside triangle. Obtuse triangle P is outside triangle.

  2. ABCis isosceles, with base AC. BDis the median to base AC. BDis an altitude of ABC. EXAMPLE 4 Prove a property of isosceles triangles Prove that the median to the base of an isosceles triangle is an altitude. SOLUTION GIVEN : PROVE :

  3. CD ADbecause BDis the median to AC. Also, BDBD. Therefore, ABD CBDby the SSS Congruence Postulate. ADB CDB because corresponding parts of are . Also, ADB and CDBare a linear pair. BDand ACintersect to form a linear pair of congruent angles, so BD ACand BDis an altitude of ABC. s Proof : Legs ABand BCof isosceles ABCare congruent. EXAMPLE 4 Prove a property of isosceles triangles

  4. SOLUTION for Examples 3 and 4 GUIDED PRACTICE 4. Copy the triangle in Example 4 and find its orthocenter.

  5. 5. WHAT IF? In Example 4, suppose you wanted to show that median BDis also an angle bisector. How would your proof be different? SOLUTION ABD CBD By SSSmaking ABD CBD which leads to BD being an angle bisector for Examples 3 and 4 GUIDED PRACTICE

  6. Triangle PQRis an isoscleles triangle and segment OQis an altitude. What else do you know about OQ? What are the coordinates of P? SOLUTION OQis also a perpendicular bisector, angle bisector, median, and altitude; (-h, 0). for Examples 3 and 4 GUIDED PRACTICE 6.

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