Coordinate Proofs

A circular irrigation system is mapped on a grid where $1 \text{ unit} = 10 \text{ feet}$. The endpoints of a diameter are located at $P(-4, 3)$ and $Q(2, -5)$. Use the midpoint and distance formulas to find the coordinates of the center of the circle and the exact length of its radius in grid units.

A community garden is shaped like a kite with vertices at $A(0, 3)$, $B(4, 6)$, $C(8, 3)$, and $D(4, -2)$. Calculate the exact total perimeter of the garden.

In triangle $ABC$ with vertices $A(-4, 1)$, $B(2, 5)$, and $C(4, -3)$, a median is drawn from vertex $B$ to side $AC$. Find the exact length of this median.

Two straight hiking trails in a national park are plotted on a map. Trail 1 passes through the points $A(-2, 1)$ and $B(2, 9)$. Trail 2 passes through the points $C(1, -3)$ and $D(4, 3)$. Prove algebraically that the two trails are parallel.

On a city planning grid, Main Street connects the points $M(0, 5)$ and $N(4, 2)$. Oak Street connects the points $O(-3, -2)$ and $P(0, 2)$. Use the slope criteria to determine if these two streets intersect at a right angle.

Prove algebraically that the triangle with vertices $A(1, 2)$, $B(4, 6)$, and $C(7, 2)$ is an isosceles triangle by calculating the lengths of its sides.

A triangular plot of land has corners at $D(0, 0)$, $E(3, 4)$, and $F(7, 1)$. Prove that the plot is a right triangle by showing that two of its sides are perpendicular.

A surveyor maps three historical landmarks at $P(-5, 2)$, $Q(1, 4)$, and $R(3, -2)$. Prove that the triangle formed by these landmarks is an isosceles right triangle.

Quadrilateral $PQRS$ has vertices $P(-1, 1)$, $Q(3, 2)$, $R(2, -2)$, and $S(-2, -3)$. Prove that $PQRS$ is a parallelogram by demonstrating algebraically that its diagonals bisect each other.

A rectangular foundation for a new building is staked out at points $A(-3, 1)$, $B(1, 4)$, $C(7, -4)$, and $D(3, -7)$ on a site map. Prove that quadrilateral $ABCD$ is a rectangle by showing it is a parallelogram and has at least one right angle.

Three vertices of a proposed parallelogram-shaped plaza are $A(-2, -1)$, $B(1, 3)$, and $C(6, 2)$. Find the coordinates of the fourth vertex $D$ (assuming the vertices are in the order $ABCD$). Then, prove algebraically whether or not the plaza is a rhombus.

A state park boundary forms a quadrilateral with vertices $W(-2, -1)$, $X(1, 3)$, $Y(6, 3)$, and $Z(3, -1)$. Prove algebraically that the park is a rhombus, and show that it is not a square.

Prove that the diagonals of the quadrilateral with vertices $K(-3, 2)$, $L(2, 4)$, $M(4, -1)$, and $N(-1, -3)$ are both perpendicular and congruent. Based on these properties, what specific type of quadrilateral is $KLMN$?

Given quadrilateral $QRST$ with vertices $Q(-5, 1)$, $R(-1, 4)$, $S(5, 4)$, and $T(9, 1)$. Prove that $QRST$ is an isosceles trapezoid by demonstrating that it has exactly one pair of parallel sides and that its non-parallel sides are congruent.

Points $A(-4, -2)$, $B(-2, 4)$, $C(4, 2)$, and $D(2, -4)$ form a quadrilateral. Prove that the midpoints of its four sides, when connected in order, form a parallelogram.

Prove that the points $A(1, 4)$, $B(4, 5)$, $C(5, 2)$, and $D(2, 1)$ form a rectangle. Next, find the midpoints of its four sides and prove that the quadrilateral formed by connecting these midpoints in order is a rhombus.

A new cell tower is built at point $T(3, 4)$ to serve a rural area. Two houses are located at $H_1(-1, 1)$ and $H_2(7, 1)$. Prove algebraically that the cell tower lies on the perpendicular bisector of the segment connecting the two houses by showing it is equidistant from both.

Find the equation of the perpendicular bisector of the segment joining $M(-2, 5)$ and $N(4, 1)$. Then, verify algebraically whether the point $P(3, 6)$ lies on this perpendicular bisector.

Two straight natural gas pipelines are laid out on a coordinate grid. Pipeline 1 connects $A(1, 8)$ and $B(7, 4)$. Pipeline 2 connects $C(1, 3)$ and $D(7, 9)$. Prove that the pipelines are neither parallel nor perpendicular, and find the exact coordinates of their intersection point.

For a triangular nature reserve with vertices $U(-2, 4)$, $V(5, 5)$, and $W(6, -2)$, find the coordinates of the circumcenter $C$ by determining the intersection of the perpendicular bisectors of two sides. Then, prove algebraically that this point is equidistant from all three vertices.

Given triangle $ABC$ with vertices $A(-20, 15)$, $B(0, -25)$, and $C(24, 7)$. Find the exact coordinates of the orthocenter by determining the equations of two altitudes and finding their intersection point. Verify algebraically that the third altitude also passes through this exact point.

A new high-speed rail station is being planned. The planners want it to be equidistant from three major tech hubs located at $A(-2, -2)$, $B(5, -1)$, and $C(-3, 5)$ on a regional map (where $1 \text{ unit} = 10 \text{ miles}$). Find the exact coordinates of the station. Then, prove algebraically that the line connecting the station to the midpoint of side $AB$ is perpendicular to $AB$.

Use a coordinate proof to show that the segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length. Place the triangle's vertices at $P(0, 2a)$, $Q(2b, 0)$, and $R(2c, 0)$ to facilitate your algebraic proof.

A drone flies over a rectangular field. Prove algebraically that for any rectangle centered at the origin with vertices $(a, b)$, $(-a, b)$, $(-a, -b)$, and $(a, -b)$, the sum of the squares of the distances from any arbitrary point $P(x, y)$ to the four vertices is equal to $4(x^2 + y^2 + a^2 + b^2)$.