Exercise 135 Page 456

Practice makes perfect

a Let's draw △ ABC using the given information.

Examining the diagram, it would appear that △ ABC is a right triangle with a right angle at ∠ A. To investigate this we must check if AB and AC are perpendicular. If they are, the slope of AB and AC should multiply to - 1. m_(AB)m_(AC)? =- 1 Let's determine the slopes of AB and AC by substituting the coordinates of the segment's endpoints into the Slope Formula.

m=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 6,6) & ( 2,3)

m_(AB)=6- 3/6- 2

SubTerm

Subtract term

m_(AB)=3/4

Let's also calculate the slope of AC.

m=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 8,-5) & ( 2,3)

m_(AC)=-5- 3/8- 2

▼

Simplify right-hand side

SubTerm

Subtract term

m_(AC)=-8/6

ReduceFrac

a/b=.a /2./.b /2.

m_(AC)=-4/3

MoveNegNumToFrac

Put minus sign in front of fraction

m_(AC)=-4/3

Now that we know the slope of the two segments, we can figure out if their product equals - 1.

m_(AB)m_(AC)? =- 1

SubstituteII

m_(AB)= 3/4, m_(AC)= - 4/3

3/4( - 4/3)? =- 1

▼

Simplify right-hand side

MultPosNeg

a(- b)=- a * b

- 3/4* 4/3 ? =- 1

MultFrac

Multiply fractions

- 12/12 ? =- 1

QuotOne

a/a=1

- 1=- 1

As we can see, AB and AC are perpendicular which means ∠ A is a right angle. Therefore, △ ABC must be a right triangle.

b Since we are reflecting the triangle in AC, only B will be affected. From Part A, we know that AB⊥ AC. Therefore, to reflect B across AC we have to extend AB across AC until AB=AB'.

Notice that △ ABC and △ AB'C share AC as a side. Therefore, these must be congruent according to the Reflexive Property of Congruence. Now we have enough information to prove that △ ABC≅ △ AB'C by the SAS (Side-Angle-Side) Congruence Theorem. Since BC and B'C are corresponding sides, they must be congruent.