Question

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate AC given \(\mathrm{AB}=15 \mathrm{~cm}\) and \(\mathrm{BC}=12 \mathrm{~cm}\)

Short answer

Answer: 9 cm

Step-by-step solution

Write down the Pythagorean theorem formula

In a right-angled triangle, the Pythagorean theorem states that: \(AB^2 = AC^2 + BC^2\)

Identify the known values and the unknown value in the equation

We are given the lengths of sides AB (15 cm) and BC (12 cm), and we need to find the length of side AC. \(15^2 = AC^2 + 12^2\)

Substitute the known values into the equation and solve for the unknown value (AC^2)

Replace AB with 15 and BC with 12: \(15^2 = AC^2 + 12^2\) \(225 = AC^2 + 144\)

Solve for AC^2

Subtract 144 from both sides of the equation: \(225 - 144 = AC^2\) \(81 = AC^2\)

Find the square root of AC^2 to get the length of side AC

Take the square root of both sides of the equation: \(\sqrt{81} = \sqrt{AC^2}\) \(9 = AC\) The length of side AC is 9 cm.

Key concepts

Right-angled Triangle

A right-angled triangle is a special type of triangle where one of the angles measures exactly 90 degrees. This right angle is critical as it allows us to use the Pythagorean theorem, a fundamental principle in geometry. A triangle like this involves three sides:

• Hypotenuse: The longest side opposite the right angle.
• Adjacent Side: The side next to the angle of interest that is not the hypotenuse.
• Opposite Side: The side opposite the angle of interest.

In \(\Delta \text{ABC}\), with C being the right angle, side AB is the hypotenuse, which is the longest side, while AC and BC are the other two sides. These sides form the foundation for applying the Pythagorean theorem which helps in calculating unknown sides of right-angled triangles.

Geometry

Geometry is the branch of mathematics that involves the study of shapes, sizes, and the properties of space. It gives us the tools we need to analyze different kinds of figures, like triangles, rectangles, and circles. Basic geometry principles allow us to understand how different shapes relate to each other and how their properties, such as angles and side lengths, can be calculated.

• Involves the study of points, lines, surfaces, solids, and their respective properties.
• Empowers us to interpret and calculate real-world dimensions and spaces.

In the context of this exercise, geometry allows us to use the Pythagorean theorem effectively. By understanding that triangle \(\Delta \text{ABC}\) is a geometric figure with a right angle at point C, we apply geometric principles to find side lengths that are not directly given. This shows how geometry serves as a practical tool for solving problems that involve sizes and spatial relationships.

Mathematics Problem Solving

Mathematics problem solving is at the heart of tackling exercises like finding the missing side in a right-angled triangle. It involves:

• Understanding the problem: Recognizing that \(\Delta \text{ABC}\) is a right-angled triangle at C.
• Planning: Deciding to use the Pythagorean theorem to find the unknown side AC.
• Executing the plan: Writing down the equation \(15^2 = AC^2 + 12^2\), substituting the known values, and solving for the unknown.
• Reviewing the results: Ensuring our calculations are correct to conclude AC = 9 cm.

The steps of this exercise illustrate the logical thinking necessary to solve mathematical problems. It highlights how structure and order make solving these problems manageable, turning abstract concepts into concrete answers.