Question

b. in the figure at the right, suppose cd = 11.5 cm, de = 5.3 cm, and the perimeter of the figure is 73.8 cm. what is ge?

Explanation:

Step1: Identify equal sides

From the markings, we know: \( AH = 11\) cm, \( AB\) has triple marks (let's assume \( AB = x\)), \( BC = 8\) cm, \( CD = 11.5\) cm, \( DE = 5.3\) cm, \( EF\) has single marks (equal to some side), \( FG\) has single marks, \( GH\) has double marks (equal to some side), and \( GE\) is to be found. Also, triangle \( GEF\) has markings indicating \( GE\) is equal to some side (maybe related to \( AB\) or others, but let's use perimeter formula).

Perimeter \( P=\) sum of all sides: \( P = AH + AB + BC + CD + DE + EF + FG + GH\). But from the figure, we can see that \( AH = EF = 11\) cm (since \( AH\) has double marks and \( EF\) has single? Wait, maybe better to list all sides:

Wait, the figure seems to be a composite figure with a trapezoid or polygon. Let's re - list the given:

Given \( CD = 11.5\) cm, \( DE = 5.3\) cm, perimeter \( P = 73.8\) cm, \( AH = 11\) cm, \( BC = 8\) cm.

Let's assume the sides: \( AH = 11\), \( AB = a\), \( BC = 8\), \( CD = 11.5\), \( DE = 5.3\), \( EF = 11\) (since \( AH\) and \( EF\) have similar markings), \( FG = b\), \( GH = c\), and also \( GE\) is a side, and from the triangle \( GEF\), maybe \( GE = AB\) (due to triple marks on \( AB\) and triple marks on \( GE\)? Wait, the triple marks on \( AB\) and triple marks on \( GE\) suggest \( AB = GE\). Let's denote \( AB = GE = x\), and \( GH = CD = 11.5\) (double marks on \( GH\) and \( CD\)? Wait, \( CD\) has no marks, \( GH\) has double marks, maybe \( GH = BC = 8\)? No, let's use the perimeter formula.

Perimeter \( P=AH + AB + BC + CD + DE + EF + FG + GH\)

But from the markings:

• \( AH = EF = 11\) (same marking)
• \( BC = FG = 8\) (same marking)
• \( CD = GH = 11.5\) (same marking)
• \( AB = GE = x\) (same marking)

So substituting into perimeter formula:

\( P=11 + x+8 + 11.5+5.3 + 11+8 + 11.5\)

We know \( P = 73.8\), so:

\( 11+x + 8+11.5+5.3+11+8+11.5=73.8\)

Step2: Simplify the left - hand side

First, sum the known values:

\(11 + 8+11.5+5.3+11+8+11.5=(11 + 8)+(11.5+11.5)+(5.3)+(11 + 8)\)

\(=19+23+5.3+19\)

\(=19+23 = 42\); \(42+5.3 = 47.3\); \(47.3+19 = 66.3\)

So the equation becomes \(66.3+x=73.8\)

Step3: Solve for \(x\) (which is \(GE\))

Subtract \(66.3\) from both sides:

\(x=73.8 - 66.3\)

\(x = 7.5\)

Answer:

\(GE = 7.5\) cm