Question

Let the vertices of a triangle \(A B C\) be \((7,9),(3,-7)\), and \((-3,3)\) then the triangle is : (a) right angled (b) equilateral (c) isosceles (d) both (a) and (c)

Short answer

Question: Given the vertices of triangle ABC, A(7, 9), B(3, -7), and C(-3, 3), determine whether the triangle is a right-angled, equilateral, isosceles, or both right-angled and isosceles triangle. Answer: (d) Both right-angled and isosceles.

Step-by-step solution

Calculate the lengths of the sides.

Using the distance formula, calculate the distances between the given vertices A(7, 9), B(3, -7), and C(-3, 3): AB: Distance between points A and B \(AB = \sqrt{(3-7)^2 + (-7-9)^2} = \sqrt{16 + 256} = \sqrt{272}\) BC: Distance between points B and C \(BC = \sqrt{(-3-3)^2 + (3+7)^2} = \sqrt{36 + 100} = \sqrt{136}\) CA: Distance between points C and A \(CA = \sqrt{(7+3)^2 + (9-3)^2} = \sqrt{100 + 36} = \sqrt{136}\)

Determine the type of triangle.

Now that we have the side lengths, let's compare them to determine the type of triangle: - If the Pythagorean theorem holds (\(a^2 + b^2 = c^2\)), the triangle will be right-angled. - If all side lengths are equal (AB = BC = CA), the triangle will be equilateral. - If two sides have equal length (AB = BC, BC = CA, or CA = AB), the triangle will be isosceles. Check Pythagorean theorem (AB is the largest side, so it should be the hypothenuse if the triangle is right-angled): \(AB^2 = 272\) \(BC^2 = 136\) \(CA^2 = 136\) \(BC^2 + CA^2 = 136 + 136 = 272 = AB^2\) Since \(AB^2 = BC^2 + CA^2\), the triangle is right-angled. Check if it is equilateral: \(AB \neq BC \neq CA\), so not equilateral. Check if it is isosceles: \(AB \neq BC\) but \(BC = CA\), so the triangle is isosceles. As the triangle is both right-angled and isosceles, the correct answer is (d) both (a) and (c).

Key concepts

Distance Formula

The Distance Formula is a fantastic tool for calculating the length between two points in a plane, which are usually represented as coordinates \(x_1, y_1\) and \(x_2, y_2\). This formula is derived from the Pythagorean Theorem and lets us find the distance by measuring the length of a straight line connecting those two points.

Here's the formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). It computes the square root of the sum of the squares of the differences in each coordinate axis.

Using this method for the vertices of triangle ABC, you calculate each side like so:

• Side AB: Use the coordinates A(7, 9) and B(3, -7) to get \(\sqrt{(3-7)^2 + (-7-9)^2}\).
• Side BC: B(3, -7) to C(-3, 3) results in \(\sqrt{(-3-3)^2 + (3+7)^2}\).
• Side CA: C(-3, 3) to A(7, 9) is \(\sqrt{(7+3)^2 + (9-3)^2}\).

By applying the Distance Formula, each side's length is quickly found, aiding in identifying the triangle's nature.

Right-Angled Triangle

A Right-Angled Triangle is special because it contains one 90-degree angle, also known as a right angle. The side opposite this angle is called the hypotenuse, and it is always the longest side.

To verify if a triangle is right-angled, you can use the Pythagorean Theorem – a powerful rule framing a relationship between the triangle's heights. According to it, for a right-angled triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other sides.

In the given problem, by comparing the squares of the side lengths: \(AB^2 = 272\), \(BC^2 = CA^2 = 136\), we see that \(AB^2 = BC^2 + CA^2\), confirming the triangle is indeed right-angled, with AB being the hypotenuse.

This tells us: if two shorter sides add up, in square form, to equal the longest side squared, you've got a right-angled triangle on your hands.

Isosceles Triangle

An Isosceles Triangle is a delightful geometry figure where two of its sides are of equal length. This equality in sides means two of the angles are also equal, providing symmetry and often, aesthetic appeal.

To check if your triangle is isosceles, you only need to compare the side lengths. If any two are equal, congratulations, your triangle is isosceles!

In our example, upon calculating the sides using the Distance Formula:

• AB: \(\sqrt{272}\)
• BC: \(\sqrt{136}\)
• CA: \(\sqrt{136}\)

The equality of sides BC and CA confirms it as isosceles. This symmetry is both a visual hallmark and functional benefit in many real-world applications.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly involving right-angled triangles. It defines the relationship between the three sides of such a triangle, stating that the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

Mathematically, it's expressed as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.

The theorem helps confirm the presence of a right angle in a triangle. In our scenario, it verifies if triangle ABC is right-angled:

• Calculate \(AB^2\), \(BC^2\), and \(CA^2\).
• Verify if \(AB^2 = BC^2 + CA^2\).

With \(272 = 136 + 136\), the theorem proves correct, showcasing AB as the hypotenuse. This classical theorem is more than just theory; it's as practical and useful today as in ancient Greece.