 As shown on the right, the vertices of a square are
(𝑥, 0), (0, 𝑦), (𝑎, 𝑏) and (𝑐, 𝑑). If 𝑎 + 𝑏 = 19 and 𝑐 + 𝑑 =
14, then 𝑥 + 𝑦 is ....

To solve this problem, let's break it down step by step.

The four vertices of a square are given as:
- \((x, 0)\)
- \((0, y)\)
- \((a, b)\)
- \((c, d)\)

We are also given the conditions:
- \(a + b = 19\)
- \(c + d = 14\)

We need to determine the value of \(x + y\).

### Step 1: Geometry of the Square
The diagonals of a square are equal in length and perpendicular to each other. The four points given form the vertices of a square, which means that the diagonals of the square should intersect at their midpoints, and the slopes of the diagonals should be perpendicular (i.e., their product should equal -1).

### Step 2: Midpoint Condition
For the diagonals to intersect at their midpoints, the midpoint of the diagonal connecting \((x, 0)\) and \((a, b)\) should be the same as the midpoint of the diagonal connecting \((0, y)\) and \((c, d)\).

- The midpoint of \((x, 0)\) and \((a, b)\) is:
\[
\left( \frac{x + a}{2}, \frac{0 + b}{2} \right) = \left( \frac{x + a}{2}, \frac{b}{2} \right)
\]

- The midpoint of \((0, y)\) and \((c, d)\) is:
\[
\left( \frac{0 + c}{2}, \frac{y + d}{2} \right) = \left( \frac{c}{2}, \frac{y + d}{2} \right)
\]

Since these midpoints must be equal, we can equate the coordinates:
1. \(\frac{x + a}{2} = \frac{c}{2}\)
2. \(\frac{b}{2} = \frac{y + d}{2}\)

### Step 3: Solve the Equations
From equation (1):
\[
x + a = c
\]
So, \(x = c - a\).

From equation (2):
\[
b = y + d
\]
So, \(y = b - d\).

### Step 4: Use Given Conditions
We are given that:
1. \(a + b = 19\)
2. \(c + d = 14\)

Now substitute into the expressions for \(x\) and \(y\):
- \(x = c - a\)
- \(y = b - d\)

To find \(x + y\), add these two expressions:
\[
x + y = (c - a) + (b - d) = (c + b) - (a + d)
\]

Using the given conditions \(a + b = 19\) and \(c + d = 14\), we can simplify this further:
\[
x + y = (c + b) - (a + d)
\]
Since \(a + d = 19 + 14 - (c + b)\),

As shown on the right, the vertices of a square are (𝑥, 0), (0, 𝑦), (𝑎, 𝑏) and (𝑐, 𝑑). If 𝑎 + 𝑏 = 19 and 𝑐 + 𝑑 = 14, then 𝑥 + 𝑦 is ....

Solve for x + y given the vertices of a square, including key algebraic steps, midpoint formula, and square properties. This problem demonstrates understanding of coordinate geometry and algebra, suitable for high school students.

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