Kyle has drawn triangle $ABC$ on a grid. Holly has started to draw an identical triangle $DEF$. We need to find the coordinates of point $F$. The coordinates of the vertices of triangle $ABC$ are $A(3, 5)$, $B(7, 6)$, and $C(4, 8)$. The coordinates of $D$ and $E$ are $D(2, 1)$ and $E(7, 1)$. Since triangle $DEF$ is identical to triangle $ABC$, we can determine the position of $F$ relative to $D$ and $E$ based on the position of $A$ relative to $B$ and $C$.

1. Problem Description

Kyle has drawn triangle $ABC$ on a grid. Holly has started to draw an identical triangle $DEF$. We need to find the coordinates of point $F$. The coordinates of the vertices of triangle $ABC$ are $A(3, 5)$, $B(7, 6)$, and $C(4, 8)$. The coordinates of $D$ and $E$ are $D(2, 1)$ and $E(7, 1)$. Since triangle $DEF$ is identical to triangle $ABC$, we can determine the position of $F$ relative to $D$ and $E$ based on the position of $A$ relative to $B$ and $C$.

2. Solution Steps

First, find the vectors $\vec{BA}$ and $\vec{BC}$.

$\vec{BA} = A - B = (3, 5) - (7, 6) = (3-7, 5-6) = (-4, -1)$.

$\vec{BE} = E - D = (7, 1) - (2, 1) = (7-2, 1-1) = (5, 0)$.

Next, determine what transformation maps $B$ to $E$. $B(7, 6)$ and $E(7, 1)$.

The x-coordinate is the same but the y coordinate changes from $6$ to $1$. It seems the identical triangle $DEF$ must be a horizontal translation and then a shift down. The corresponding points are as follows: $A$ to $D$, $B$ to $E$ and $C$ to $F$.

Therefore, vector $\vec{AB}$ is equal to vector $\vec{DE}$.

$B - A = (7, 6) - (3, 5) = (4, 1)$.

$E - D = (7, 1) - (2, 1) = (5, 0)$.

Since the vector $DE$ should match vector $AB$ it seems.

The difference in the vector $\vec{DE}$ and $\vec{AB}$ indicates a slight stretch and rotation. Since they are supposed to be "identical" this would mean that point $A$ would be the new point $D$ point $B$ would be point $E$ and point $C$ would be point $F$.

So $\vec{AC} = C - A = (4, 8) - (3, 5) = (1, 3)$.

We need to add this vector to $D$.

$F = D + \vec{AC} = (2, 1) + (1, 3) = (3, 4)$.

3. Final Answer

(3, 4)