use the diagram shown. find eg.
eg=\boxed{}
(simplify your answer.)

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Accepted Answer

# Explanation:
## Step1: Identify the theorem
We use the Angle Bisector Theorem and the property of equal - distance from a point on the angle bisector to the sides of the angle. Also, we can use the concept of similar triangles or the property of the in - center (or a point equidistant from the sides of a triangle) and the ratio of the lengths.

We know that $GD = 11$, $BG = 14$, and $BC=42$. Let's consider the triangles $BEG$ and $BDG$ and the fact that the ratio of the segments of the side $BC$ and the corresponding distances from $G$ to the sides $AB$ and $BC$ can be related.

First, we note that the distance from $G$ to $AB$ is $GD = 11$ and the distance from $G$ to $BC$ is $EG$. Also, the length of $BE$ and $BC$ are related. Wait, actually, we can use the property of the angle bisector and the ratio of the sides.

We know that the point $G$ is equidistant from $AB$ and $AC$ (since $GD\perp AB$, $GF\perp AC$) and also from $AB$ and $BC$ (since $GD\perp AB$, $EG\perp BC$).

We can use the ratio of the lengths of the segments of the side $BC$ and the lengths of the perpendiculars from $G$ to the sides.

We know that the ratio of $BE$ to $BC$ is equal to the ratio of $GD$ to $EG$? Wait, no. Let's think about the area or the similar triangles.

Alternatively, we can use the concept of the angle bisector and the ratio of the adjacent sides.

We know that $GD$ is perpendicular to $AB$, $EG$ is perpendicular to $BC$, and $BG$ is the angle bisector of $\angle ABC$. By the Angle - Bisector Theorem, the ratio of the distances from a point on the angle bisector to the two sides of the angle is equal to the ratio of the adjacent sides. Wait, the Angle - Bisector Theorem states that $\frac{AB}{BC}=\frac{AD}{DC}$, but here we are dealing with the distances from $G$ to the sides.

Wait, actually, the distance from $G$ to $AB$ is $GD = 11$ and the distance from $G$ to $BC$ is $EG$. Also, we can consider the triangles $BDG$ and $BEG$. Since $\angle BDG=\angle BEG = 90^{\circ}$ and $\angle DBG=\angle EBG$ (because $BG$ is the angle bisector), the triangles $BDG$ and $BEG$ are similar by the AA (Angle - Angle) similarity criterion.

In similar triangles, the ratio of corresponding sides is equal. But we can also use the ratio of the lengths of $BE$ and $BD$ and $EG$ and $GD$. Wait, we know that $BC = 42$, and let's assume that $BE$ is a part of $BC$. Wait, maybe we can use the ratio of the lengths of the perpendiculars.

Wait, another approach: The area of $\triangle ABG$ can be calculated as $\frac{1}{2}\times AB\times GD$ and the area of $\triangle CBG$ can be calculated as $\frac{1}{2}\times BC\times EG$. Also, the ratio of the areas of $\triangle ABG$ and $\triangle CBG$ is equal to the ratio of $AB$ to $BC$ (if they share the same vertex $G$)? No, actually, if $BG$ is the angle bisector, then $\frac{[ABG]}{[CBG]}=\frac{AB}{BC}$ (by the formula for the area of a triangle $\frac{1}{2}ab\sin C$, where $a$ and $b$ are two sides and $C$ is the included angle). But we also know that $[ABG]=\frac{1}{2}\times AB\times GD$ and $[CBG]=\frac{1}{2}\times BC\times EG$. So $\frac{\frac{1}{2}\times AB\times GD}{\frac{1}{2}\times BC\times EG}=\frac{AB}{BC}$, which simplifies to $\frac{GD}{EG} = 1$, which is not correct. So that approach is wrong.

Wait, let's look at the lengths. We know that $GD = 11$, and we can use the ratio of the segments. Let's assume that the length of $BE$ is $x$ and $EC=42 - x$. But we also know that $GD$ and $EG$ are the distances from $G$ to $AB$ and $BC$ respectively.

Wait, another way: The point $G$ is the in - center? No, the in - center is equidistant from all three sides, but here we have $GD = 11$, and we need to find $EG$. Wait, maybe we can use the ratio of the lengths of $BG$ and the other segments.

Wait, let's consider the right triangles $BDG$ and $BEG$. $\angle BDG=\angle BEG = 90^{\circ}$, $\angle DBG=\angle EBG$, so $\triangle BDG\sim\triangle BEG$ (AA similarity). Then $\frac{GD}{EG}=\frac{BD}{BE}$. But we don't know $BD$ and $BE$.

Wait, maybe we can use the concept of the angle bisector and the ratio of the adjacent sides. Wait, the length of $BC = 42$, and let's assume that the ratio of $AB$ to $BC$ is related to the ratio of $GD$ to $EG$. But we know that $GD = 11$, and we can use the fact that the product of the segments of the angle bisector and the sides.

Wait, I think I made a mistake. Let's look at the diagram again. We see that $GD = 11$, $BG = 14$, and $BC = 42$. We can use the property of the angle bisector and the ratio of the perpendiculars.

The key idea is that the distance from a point on the angle bisector to the two sides of the angle is proportional to the adjacent sides. Wait, no, the distance from the point to the two sides is equal? No, that's only if the point is on the angle bisector, the distance to the two sides is equal. Wait, yes! If $BG$ is the angle bisector of $\angle ABC$, then the distance from $G$ to $AB$ (which is $GD$) is equal to the distance from $G$ to $BC$ (which is $EG$)? No, that's not right. Wait, no, the distance from a point on the angle bisector to the two sides of the angle is equal. So if $BG$ bisects $\angle ABC$, and $GD\perp AB$, $EG\perp BC$, then $GD = EG$. But $GD = 11$, so $EG = 11$? But that seems too simple. Wait, no, maybe I misread the diagram.

Wait, no, maybe the diagram has $BE$ and $BC$ such that $BE$ is a part of $BC$. Wait, the length of $BC$ is $42$, and maybe we can use the ratio of the lengths of $BG$ and the other segments. Wait, let's calculate using the geometric mean or the Pythagorean theorem.

In right triangle $BDG$, $BD=\sqrt{BG^{2}-GD^{2}}=\sqrt{14^{2}-11^{2}}=\sqrt{196 - 121}=\sqrt{75}=5\sqrt{3}$. But that may not be helpful.

Wait, another approach: The point $G$ is the in - center, and the lengths of the perpendiculars from $G$ to the sides are equal. But $GD$ is perpendicular to $AB$, $EG$ is perpendicular to $BC$, and $GF$ is perpendicular to $AC$. So $GD = EG=GF$. But $GD = 11$, so $EG = 11$? But that seems conflicting with the length of $BC = 42$. Wait, maybe the diagram is such that $BE$ is not the entire $BC$. Wait, no, the problem is to find $EG$, and if $G$ is on the angle bisector of $\angle ABC$, then the distance from $G$ to $AB$ ( $GD$) and the distance from $G$ to $BC$ ( $EG$) should be equal. So $EG=GD = 11$? But that seems too simple. Wait, maybe I made a mistake.

Wait, let's check again. The Angle - Bisector Theorem for distances: A point on the angle bisector is equidistant from the two sides of the angle. So since $BG$ bisects $\angle ABC$, and $GD\perp AB$, $EG\perp BC$, then $GD = EG$. Since $GD = 11$, then $EG = 11$.

## Step2: Confirm the result
Since $G$ lies on the angle bisector of $\angle ABC$ (as indicated by the equal angle marks), and $GD$ and $EG$ are the perpendicular distances from $G$ to $AB$ and $BC$ respectively, by the property of the angle bisector, these two distances are equal. So $EG=GD = 11$.

# Answer:
$11$

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