Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 27

Chapter 3, Problem 27

Determine whether the three points are the vertices of a right triangle. (-4,3),(2,5),(-1,-6)

Verified step by step guidance

Identify the three points given: A(-4, 3), B(2, 5), and C(-1, -6).

Calculate the distance between each pair of points using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Find the lengths of the sides: $AB$, $BC$, and $AC$ by substituting the coordinates of the points into the distance formula.

Check if the triangle is right-angled by verifying the Pythagorean theorem: see if the square of the longest side equals the sum of the squares of the other two sides, i.e., $c^2 = a^2 + b^2$.

Conclude whether the points form a right triangle based on the Pythagorean theorem check.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distance Formula

The distance formula calculates the length between two points in the coordinate plane using their coordinates. It is derived from the Pythagorean theorem and is given by √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the lengths of the sides of the triangle formed by the given points.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. To determine if a triangle is right-angled, check if the side lengths satisfy a² + b² = c², where c is the longest side.

Identifying Right Triangles Using Coordinates

To verify if three points form a right triangle, calculate the distances between each pair of points to find side lengths. Then apply the Pythagorean theorem to these lengths. If the relationship holds for any combination, the points form a right triangle.

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Graphs and Coordinates - Example

Related Practice

Textbook Question

Determine whether the three points are the vertices of a right triangle. See Example 3.

$(- 2, - 5), (1, 7), (3, 15)$

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Give the center and radius of the circle represented by each equation. x²+y²+6x+8y+9=0

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Graph each line. Give the domain and range. x = -4

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Determine whether each equation defines y as a function of x. y = ±√(x-2)

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Textbook Question

Determine whether each relation defines a function, and give the domain and range.

Textbook Question

Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. m=5, b=15