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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

5. Isosceles and Equilateral Triangles

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Exercise 38 Page 380

Consider the Converse of Isosceles Triangle Theorem.

x=-8, x=3

Practice makes perfect

Consider the given triangle.

We want to find the value of x. To do so, we will start by identifying the type of triangle. Let's recall the classification of triangles.

Classification of Triangles
Scalene Triangle	A scalene triangle is a triangle in which all three sides have different lengths.
Isosceles Triangle	An isosceles triangle is a triangle that has two congruent sides and two base angles with the same measure.
Equilateral Triangle	An equilateral triangle is a triangle in which all the sides are congruent.
Acute Triangle	An acute triangle is a triangle where all angles are less than 90^(∘) or π2.
Obtuse Triangle	An obtuse triangle is a triangle with exactly one an angle whose measure is greater than 90^(∘) or π2.
Right Triangle	A right triangle is a specific type of triangle that contains one angle of 90^(∘).

Since the given triangle has two congruent angles, the given triangle is an isosceles triangle. Then, we can consider the Converse of Isosceles Triangle Theorem to find x.

Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

By this theorem, we can conclude that x^2+5x=24. Then, we can use this equation to find x.

x^2+5x=24

LHS-24=RHS-24

x^2+5x-24=0

Write as a difference

x^2+8x-3x-24=0

Factor out x

x(x+8)-3x-24=0

Factor out -3

x(x+8)-3(x+8)=0

Factor out (x+8)

(x+8)(x-3)=0

Zero Property of Multiplication

lcx+8=0 & (I) x-3=0 & (II)

(I): LHS-8=RHS-8

lx=-8 x-3=0

(II): LHS+3=RHS+3

lx=-8 x=3

As a result, x can be either -8 or 3.

Subchapter links

Exercises p.378-382