Problem 46 - Algebra 1: Concepts and Skills

Question

MULTIPLE CHOICE Given the lengths of the three sides of a triangle, determine which triangle is not a right triangle. $$ A)a=9, b=40, c=41 $$ $$ B)a=3, b=4, c=5 $$ $$ C)a=7, b=24, c=25 $$ $$ D)a=10, b=49, c=50 $$

Step-by-Step Solution

Answer

The triangle that is not a right triangle is D, having sides a = 10, b = 49, c = 50.

Step 1: Check Triangle A

For Triangle A, we calculate if \( c^2 \) equals \( a^2 + b^2 \). Using the given sides: \(41^2 = 9^2 + 40^2\) or \(1681 = 81 + 1600\). See that this equation is true, implying that Triangle A is a right triangle.

Step 2: Check Triangle B

Doing the same with Triangle B, we calculate if \(5^2 = 3^2 + 4^2\). The equation results as \(25 = 9 + 16\), and the equation is also true. Hence, Triangle B is also a right triangle.

Step 3: Check Triangle C

We can continue with Triangle C, we calculate if \(25^2 = 7^2 + 24^2\). The equation results as \(625 = 49 + 576\), which satisfies Pythagorean theorem. Hence, Triangle C is a right triangle.

Step 4: Check Triangle D

Lastly, we evaluate Triangle D: \(50^2 = 10^2 + 49^2\) simplifies to \(2500 = 100 + 2401\). This equation is false, implying that Triangle D is not a right triangle.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that always has one angle measuring exactly 90 degrees. This unique characteristic also dictates certain properties for the sides of the triangle, noted as the Pythagorean Theorem.

• In a right triangle, the square of the length of the longest side, called the hypotenuse, is equal to the sum of the squares of the lengths of the other two sides.
• The formula for this relationship is expressed as: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
• This theorem helps determine whether a triangle is a right triangle by checking if the equation holds true for given side lengths.

For example, in the step by step solution for the right triangles in this exercise, each triangle satisfying this equation is confirmed to be a right triangle. Confirming whether a triangle is right can help solve various geometry problems more efficiently.

Triangle Inequality

Triangle inequality is a fundamental rule in geometry, necessary for understanding what makes a set of three line segments a triangle.

• It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• This principle ensures that the two sides can meet to form a closed shape, which is essential for defining a triangle.
• For example, among the sets of sides given in the exercise, all can form a triangle because they satisfy this inequality rule.

The triangle inequality is critical during the initial assessment when determining whether a set of given sides can even form a triangle in the first place. It ensures that the properties of the triangle are physically possible. Without passing the triangle inequality, the set of sides cannot form a valid triangle.

Geometry

Geometry is a broad mathematical field that addresses the properties and relations of points, lines, surfaces, and solids. This particular exercise dives into how geometry relates to triangles, specifically focusing on right triangles.

• Understanding right triangles and their properties is essential because they frequently appear in both theoretical and applied geometry.
• The Pythagorean theorem is deeply rooted in the branch of geometry known as Euclidean geometry, which deals with figures on a flat plane.
• Analyzing triangles within the realm of geometry not only helps in recognizing patterns and relationships but also assists in practical applications like construction or design.

The exercise showcases how geometry and algebra intersect, as solving for right triangles involves applying algebraic methods to geometric principles. This interconnectedness is a key aspect of geometry that makes it so valuable in various real-world scenarios.