Question

question 3
1 pts
what is the value of x?
32
68
41
24

Explanation:

Step1: Identify Similar Triangles

From the diagram, we can see that \( \triangle EGD \sim \triangle EHN \) by the Angle - Angle (AA) similarity criterion (since \( \angle GED=\angle HEN \) and \( \angle GDE=\angle HNE \) or by the property of angle - bisector and proportional sides). For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{EG}{EH}=\frac{GD}{HN} \).

We know that \( EG = 44.8\space mm \), \( EH = 56\space mm \), \( GD=(x + 4)\space mm \), and \( HN = 35\space mm \).

Step2: Set Up the Proportion

Substitute the values into the proportion:
\( \frac{44.8}{56}=\frac{x + 4}{35} \)

Step3: Cross - Multiply

Cross - multiply to solve for \( x+4 \):
\( 56\times(x + 4)=44.8\times35 \)
First, calculate \( 44.8\times35 \): \( 44.8\times35 = 1568 \)
Then, divide both sides by 56: \( x + 4=\frac{1568}{56} \)
\( \frac{1568}{56}=28 \), so \( x + 4 = 28 \)? Wait, no, wait. Wait, maybe I mixed up the sides. Wait, actually, the correct proportion should be \( \frac{EG}{EH}=\frac{GD}{HN} \)? Wait, no, let's re - examine. Wait, \( EG = 44.8 \), \( EH = 56 \), \( GD=(x + 4) \), \( HN = 35 \). Wait, maybe the correct proportion is \( \frac{EG}{EH}=\frac{GD}{HN} \)? Wait, no, let's check the lengths again. Wait, \( EG = 44.8 \), \( EH = 56 \), \( GD=(x + 4) \), \( HN = 35 \). Wait, \( \frac{44.8}{56}=\frac{x + 4}{35} \). Let's solve this:

\( 56\times(x + 4)=44.8\times35 \)

\( 56x+224 = 1568 \)

\( 56x=1568 - 224 \)

\( 56x = 1344 \)

\( x=\frac{1344}{56}=24 \)? Wait, no, that's not right. Wait, maybe the proportion is \( \frac{EG}{EH}=\frac{GD}{HN} \) is incorrect. Wait, maybe the correct proportion is \( \frac{EG}{EH}=\frac{GD}{HN} \)? Wait, no, let's look at the sides again. Wait, \( EG = 44.8 \), \( EH = 56 \), \( GD=(x + 4) \), \( DN = 35 \)? Wait, no, the segment \( HD = 35 \), and \( GD=(x + 4) \). Wait, maybe the triangles are \( \triangle EGD \) and \( \triangle EHD \)? No, the angle at \( E \) is common? Wait, no, the angle at \( E \) is bisected? Wait, the diagram shows that \( \angle GE D=\angle DEH \), so by the Angle - Bisector Theorem, which states that an angle bisector in a triangle divides the opposite side into segments proportional to the adjacent sides. So, in \( \triangle EGH \), \( ED \) is the angle bisector of \( \angle GEH \), so \( \frac{EG}{EH}=\frac{GD}{DH} \)

Ah, that's the mistake! The Angle - Bisector Theorem: In a triangle, the angle bisector divides the opposite side into segments proportional to the adjacent sides. So, \( \frac{EG}{EH}=\frac{GD}{DH} \)

So, \( EG = 44.8\space mm \), \( EH = 56\space mm \), \( GD=(x + 4)\space mm \), \( DH = 35\space mm \)

So, \( \frac{44.8}{56}=\frac{x + 4}{35} \)

Wait, no, \( \frac{EG}{EH}=\frac{GD}{DH} \), so \( \frac{44.8}{56}=\frac{x + 4}{35} \)

Cross - multiply: \( 56\times(x + 4)=44.8\times35 \)

\( 56x+224 = 1568 \)

\( 56x=1568 - 224=1344 \)

\( x=\frac{1344}{56}=24 \)? But that gives \( x = 24 \), but let's check again. Wait, \( 44.8\div56 = 0.8 \), and \( 35\times0.8 = 28 \), so \( x + 4=28 \), so \( x=24 \). Wait, that's correct.

Wait, let's verify: If \( x = 24 \), then \( GD=24 + 4=28\space mm \). Then \( \frac{EG}{EH}=\frac{44.8}{56}=0.8 \), and \( \frac{GD}{DH}=\frac{28}{35}=0.8 \). So the ratios are equal, which satisfies the Angle - Bisector Theorem.

Answer:

\( x = 24 \)