Problem 22 - Algebra 2

Question

GEOMETRY Quadrilateral RSTV has vertices \(R(-4,6), S(4,5), T(6,3),\) and \(V(5,-8) .\) Find the perimeter of the quadrilateral.

Step-by-Step Solution

Answer

The perimeter is approximately 44.59 units.

Step 1: Calculate the Distance Between R and S

We will use the distance formula to calculate the distance between points \(R(-4, 6)\) and \(S(4, 5)\). The distance formula is \ \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Apply the coordinates: \ \( RS = \sqrt{(4 - (-4))^2 + (5 - 6)^2} = \sqrt{(8)^2 + (-1)^2} = \sqrt{64 + 1} = \sqrt{65} \).

Step 2: Calculate the Distance Between S and T

Using the distance formula, calculate the distance between \(S(4, 5)\) and \(T(6, 3)\): \ \( ST = \sqrt{(6 - 4)^2 + (3 - 5)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \).

Step 3: Calculate the Distance Between T and V

Calculate the distance between \(T(6, 3)\) and \(V(5, -8)\): \ \( TV = \sqrt{(5 - 6)^2 + (-8 - 3)^2} = \sqrt{(-1)^2 + (-11)^2} = \sqrt{1 + 121} = \sqrt{122} \).

Step 4: Calculate the Distance Between V and R

Calculate the distance between \(V(5, -8)\) and \(R(-4, 6)\): \ \( VR = \sqrt{(-4 - 5)^2 + (6 + 8)^2} = \sqrt{(-9)^2 + 14^2} = \sqrt{81 + 196} = \sqrt{277} \).

Step 5: Find the Perimeter of Quadrilateral RSTV

Add up all the distances calculated in the previous steps to find the perimeter: \ \( \text{Perimeter} = RS + ST + TV + VR = \sqrt{65} + \sqrt{8} + \sqrt{122} + \sqrt{277} \). Use a calculator to sum these values numerically.

Key Concepts

Distance Formula

To find the perimeter of any shape with vertices placed on a coordinate plane, we first need to measure the distances between each pair of adjacent points. This is where the distance formula comes in handy. The distance formula, which is derived from the Pythagorean theorem, helps us calculate the straight line distance between two points in a plane and is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]For example, to find the distance between points \(R(-4, 6)\) and \(S(4, 5)\), you plug their coordinates into the formula.

• Subtract the x-coordinates: \((x_2 - x_1) = (4 - (-4)) = 8\)
• Subtract the y-coordinates: \((y_2 - y_1) = (5 - 6) = -1\)
• Square each result: \(8^2 = 64\) and \((-1)^2 = 1\)
• Add and take the square root: \(\sqrt{64 + 1} = \sqrt{65}\)

Use these steps for each pair of adjacent vertices to find all side lengths.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics that helps us represent and analyze geometric figures using a coordinate system. This approach allows us to apply algebraic techniques to solve geometric problems. Placing geometric figures like quadrilaterals in a coordinate plane simplifies calculations of distances, angles, and other properties. In the case of a quadrilateral with given vertices, other than determining side lengths, one can also use coordinate geometry to:

• Determine whether the figure is a special kind of quadrilateral (rectangle, square, etc.) by checking slopes and lengths.
• Calculate the area using formulas like the trapezoid rule.
• Verify symmetry and determine points of interest such as midpoints or centroid.

This method is powerful, not just for classrooms or homework, but as a fundamental practice in fields like engineering, architecture, and computer graphics.

Quadrilaterals

A quadrilateral is a four-sided polygon with four vertices and four edges. The vertices connect pairwise to form four angles. Different types of quadrilaterals include squares, rectangles, trapezoids, and rhombuses, each with its own specific properties. For example, it's essential to understand that:

• A square has all sides equal and every angle is a right angle.
• A rectangle has opposite sides equal, and all angles are right angles.
• A rhombus has all sides equal, but not necessarily 90-degree angles.
• A trapezoid has at least one pair of parallel sides.

Quadrilaterals on a coordinate plane like RSTV, even if irregular, require calculating the perimeter by summing the lengths of all sides, which leaves us with a better understanding of their form and dimensions.