Question

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate \(\mathrm{AB}\) given \(\mathrm{AC}=9 \mathrm{~cm}\) and \(\mathrm{BC}=15 \mathrm{~cm}\)

Short answer

Answer: The length of the hypotenuse AB is approximately 17.49 cm.

Step-by-step solution

Recall the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs of length a and b, and a hypotenuse of length c: \(a^2+b^2=c^2\). We will use this theorem to find the length of the hypotenuse AB.

Identify the given lengths

We are given that AC = 9 cm and BC = 15 cm. We can assign AC = a and BC = b, so we have a = 9 and b = 15.

Apply the Pythagorean theorem

Using the Pythagorean theorem, we will plug in the values of a and b into the formula and solve for c: \(a^2+b^2=c^2\) \((9)^2+(15)^2=c^2\)

Calculate the squares of a and b

Now, we will calculate the squares of a and b and then add the two values together: \((9)^2 = 81\) \((15)^2 = 225\) \(81 + 225 = 306\) So, the equation becomes: \(306 = c^2\)

Find the length of the hypotenuse

To find the length of the hypotenuse (c), we will take the square root of both sides of the equation: \(\sqrt{306} = c\) Approximating the square root of 306, we get: \(c \approx 17.49\) So, the length of the hypotenuse AB is approximately 17.49 cm.

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees, known as a right angle. This characteristic defines the right triangle and gives rise to important properties and relationships among its sides. The side opposite the right angle is called the hypotenuse, which is always the triangle's longest side. The other two sides are known as the legs of the triangle.

In a right triangle, the legs and the hypotenuse have a special relationship described by the Pythagorean theorem, which we will delve into later. Understanding the nature of a right triangle helps in problem solving as it allows us to apply specific mathematical properties and formulas that apply only to right triangles. This makes right triangles incredibly useful in fields such as geometry, trigonometry, and many real-world applications, like architecture and engineering.

Hypotenuse Calculation

When solving problems involving right triangles, calculating the length of the hypotenuse is often a key step. The hypotenuse is crucial because it is the longest side, stretching across the right angle. The Pythagorean theorem is the tool we use to find the hypotenuse length when the other two side lengths are known.

According to this theorem, the square of the hypotenuse's length (\( c \)) is equal to the sum of the squares of the other two legs (\( a \) and \( b \)). The formula is written as follows:

• \( a^2 + b^2 = c^2 \)

This equation allows us to solve for the unknown side of the triangle. In the case of the exercise, knowing that one leg measures 9 cm and the other measures 15 cm, we substitute these values into the theorem. The calculations reveal that the hypotenuse length is approximately 17.49 cm. This illustrates the simple yet powerful nature of the Pythagorean theorem in solving practical mathematical problems.

Mathematical Problem Solving

Solving mathematical problems like the one given in the exercise involves a step-by-step methodical approach. Here's how you can think through such problems:

• Identify the problem type: Recognize that this is a right triangle problem, leading you to the appropriate formulas and methods.
• Gather known information: Note down all given side lengths and any known angles. This helps in substituting values into the correct formula.
• Apply the correct mathematical principle: Use the Pythagorean theorem since it directly applies to right triangles with known leg lengths.
• Calculate with care: Perform calculations, such as squaring each leg's length and then adding them to find the square of the hypotenuse.
• Solve for the unknown: Take the square root to find the hypotenuse length or any other side if applicable.

Applying these problem-solving steps ensures that you can effectively tackle similar exercises in the future. This structured approach is beneficial in breaking down complex problems into manageable parts, ultimately enhancing your mathematical reasoning and skills.