College Algebra Examples: Applications of the Distance and Midpoint Formulas

Summary

This educational video script explores the application of distance and midpoint formulas through geometric examples. It begins with a problem involving a triangle, calculating side lengths and confirming it's a right triangle using the Pythagorean theorem. The script then demonstrates calculating the area of the triangle using geometric manipulation. The second example applies these formulas to a baseball scenario, determining the distance a shortstop must throw to home plate. The script effectively illustrates the practical use of mathematical concepts in real-world situations.

Takeaways

• 📚 The video discusses the application of distance and midpoint formulas through examples.
• 📐 The first example involves finding the lengths of the sides of a triangle with given vertex coordinates.
• 📈 The distance formula is used to calculate the lengths of the sides of the triangle ABC.
• 🔍 The coordinates of points A, B, and C are given in terms of a variable 'a', which is used in calculations.
• 📉 The lengths of the sides are found to be the square root of 20, 5, and the square root of 5 respectively.
• 🔺 Part B of the problem asks to verify if the triangle is a right triangle using the Pythagorean theorem.
• 📐 The triangle is confirmed to be a right triangle as the sum of the squares of the two shorter sides equals the square of the longest side.
• 📏 The area of the right triangle is calculated using the formula (1/2) * base * height, resulting in an area of 5.
• 🏀 The second problem involves a baseball diamond and a short stop throwing to home plate.
• 📍 The baseball diamond is modeled as a square with home plate at the origin and the bases at specific coordinates.
• 🤸‍♂️ The midpoint formula is used to find the position of the short stop, who is halfway between second and third base.
• 📏 The distance from the short stop to home plate is calculated using the distance formula, resulting in approximately 100.6 feet.

Q & A

What are the coordinates of point A in the first example problem?

The coordinates of point A are (-2a, 1).

What is the formula used to calculate the distance between two points?

The distance formula is used, which is the square root of the difference in the x-coordinates squared plus the difference in the y-coordinates squared.

How is the length of each side of triangle ABC found?

The length of each side is found by applying the distance formula to the coordinates of the respective points.

What is the length of side AB in triangle ABC?

The length of side AB is the square root of 20.

How can you verify if a triangle is a right triangle?

You can verify if a triangle is a right triangle by checking if the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side (the hypotenuse).

What is the relationship between the sides of a right triangle in terms of their lengths?

In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

What is the area of the right triangle ABC?

The area of the right triangle ABC is 5 square units, calculated by taking half the product of the lengths of the two legs (base and height).

How is the midpoint of a segment found?

The midpoint of a segment is found by averaging the x-coordinates and the y-coordinates of the endpoints of the segment.

What is the scenario described in the second problem involving a baseball diamond?

The second problem involves a shortstop standing halfway between second and third base on a baseball diamond and needing to throw the ball to home plate, with the goal of determining the distance from the shortstop to home plate.

How is the distance from the shortstop to home plate calculated in the second problem?

The distance is calculated using the distance formula with the coordinates of the shortstop (45, 90) and home plate (0, 0).

What is the approximate distance the shortstop has to throw the ball to reach home plate?

The shortstop has to throw the ball approximately 100.6 feet to reach home plate.