**Answer:**

Square; 68.0; 289.0

**Step-by-step explanation:**

**1. Shape
**

Your shape is plotted in the graph below.

It looks like a square.

**2. Proof
**

To prove that the object is a square , we must show that **adjacent sides are perpendicular** and that the **diagonals are perpendicular**.

**(a) Sides
**

If the slopes of thee lines are negative reciprocals, the lines are perpendicular, and the angles are 90 °.

The formula for the slope (rate of change) m of a straight line is

m = Δy/Δx = (y₂ -y₁)/(x₂ - x₁)

**(i) AB
**

m = [-11 - (-19)]/(25 - 10) = (-11 +19)/15 = 8/15

**(ii) BC
**

m = [-4 - (-19)]/(2 - 10) = (-4 +19)/(-8) = -15/(-8)= -15/8

The slopes of AB and BC are negative reciprocals, so ∠B = 90 °.

**(iii) CD
**

m = [4 - (-4)]/(17 - 2) = (4 +4)/15 = 8/15

The slopes of BC and CD are negative reciprocals, so ∠C = 90 °.

**(iv) AD
**

m = [4 - (-11)]/(17 - 25) = (4 +11)/(-8) = 15/(-8)= -15/8

The slopes of CD and AD are negative reciprocals, so ∠D = 90 °.

Also, the slopes of AD and AB are negative reciprocals, so ∠ = 90 °.

**All four angles are 90°.
**

**(b)Diagonals
**

**AC**: m = [-11 - (-4)]/(25- 2) = (-11 +4)/23 = -7/23

**BD**: m = [4 - (-19)]/(17 - 10) = (4 +19)/7 = 23/7

The slopes are negative reciprocals, so the **diagonals are perpendicular.
**

The adjacent sides of quadrilateral ABCD have negative reciprocal slopes, 8/15 and -15/8, so, they form right angles and the slopes of the diagonals are negative reciprocals, -7/23 and 23/7, so they form right angles. Therefore the quadrilateral is a **square**.

**3. Perimeter
**

The formula for the** perimeter of a square** is

P = 4s

where s is the side length

AB² = 8² + 15²

s² = 64 + 225

s² = **289.0 **

s = 17.0

P = 4 × 17.0 = **68.0
**

**4. Area
**

The formula for the **area of a square** is

A = s²

A =** 289.0
**