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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

4. Proving Triangles Congruent-ASA, AAS

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Exercise 31 Page 371

Calculate the lengths of the sides of the triangles using the Distance Formula.

Yes, see solution.

Practice makes perfect

To see whether △ ABC and △ XYZ are congruent or not, let's find the lengths of the sides.

Since we know the coordinates of the vertices, we can use the Distance Formula. Let's start with finding AB.

AB=sqrt((x_B-x_A)^2+(y_B-y_A)^2)

SubstitutePoints

Substitute ( 6,4) & ( 1,- 6)

AB=sqrt(( 1- 6)^2+( - 6- 4)^2)

▼

Simplify right-hand side

Subtract terms

AB=sqrt((- 5)^2+(- 10)^2)

NegBaseToPosBase

(- a)^2=a^2

AB=sqrt(5^2+10^2)

Calculate power

AB=sqrt(25+100)

Add terms

AB=sqrt(125)

We can find the lengths of the other sides in a similar way. Before we find the lengths, notice that the exercise asks about the congruence in a specific order. △ ABC? ≅△ XYZ This means that we need to check the lengths between the corresponding sides to see if each segment is congruent.

Corresponding Sides	Distance Formula	Result
AB and XY	sqrt((1-6)^2+(- 6-4)^2) ? = sqrt((5-0)^2+(- 3-7)^2)	sqrt(125)= sqrt(125)
BC and YZ	sqrt((- 9-1)^2+(5-(- 6))^2) ? = sqrt((15-5)^2+(8-(- 3))^2)	sqrt(221)= sqrt(221)
CA and ZX	sqrt((6-(- 9))^2+(4-5)^2) ? = sqrt((0-15)^2+(7-8)^2)	sqrt(226)= sqrt(226)

Since all three side pairs are congruent, the Side-Side-Side (SSS) Congruence Postulate guarantees that the triangles are congruent. △ ABC ≅△ XYZ

Subchapter links

Exercises p.367-371