Question

note: image not drawn to scale. what is the value of x? enter your answer in the box. x = box (the image shows a triangle with points e, g, h; eg is 49 cm, gd is 35 cm, eh is 58.8 cm, dh is (x + 1) cm, and ed is a segment inside the triangle)

Explanation:

Step1: Identify Similar Triangles

From the diagram, we can infer that triangle \( \triangle EGD \) and triangle \( \triangle EHG \) are similar (by the Angle - Bisector Theorem or by the Side - Angle - Side similarity criterion, as \( \angle GED=\angle DEH \) and the sides around the angle are proportional). So, the ratios of corresponding sides are equal. That is, \( \frac{EG}{EH}=\frac{GD}{DH} \).

We know that \( EG = 49\space cm \), \( EH=58.8\space cm \), \( GD = 35\space cm \), and \( DH=(x + 1)\space cm \).

Step2: Set up the Proportion

Substitute the known values into the proportion:
\( \frac{49}{58.8}=\frac{35}{x + 1} \)

Step3: Cross - Multiply

Cross - multiplying gives us \( 49\times(x + 1)=58.8\times35 \)

First, calculate \( 58.8\times35 \):
\( 58.8\times35=(60 - 1.2)\times35=60\times35-1.2\times35 = 2100-42=2058 \)

So, the equation becomes \( 49(x + 1)=2058 \)

Step4: Solve for \( x+1 \)

Divide both sides of the equation by 49:
\( x + 1=\frac{2058}{49} \)
\( \frac{2058}{49}=42 \) (since \( 49\times42 = 2058 \))

Step5: Solve for \( x \)

Subtract 1 from both sides:
\( x=42 - 1=41 \)

Answer:

\( 41 \)