Question

Points \(A(6,6), B(2,3)\) and \(C(4,7)\) are the vertices of a triangle which is : (a) right angled (b) acute angled (c) obtuse angled (d) none of these

Short answer

a^2+b^2>c^2$, where c is the length of the longest side, then the triangle is acute-angled. If $a^2+b^2 BC^2 = 20\), which shows it is an acute-angled triangle since only one condition must be satisfied. Therefore, the triangle is (b) acute-angled.

Step-by-step solution

Find the lengths of the sides AB, BC, and AC

We will use the distance formula to find the lengths of the sides: \(AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\), so \(AB=\sqrt{(2-6)^2+(3-6)^2} = \sqrt{(-4)^2+(-3)^2} = \sqrt{16+9} = \sqrt{25}=5\). Likewise, \(BC=\sqrt{(4-2)^2+(7-3)^2} = \sqrt{2^2+4^2} = \sqrt{4+16} = \sqrt{20}\), and \(AC=\sqrt{(6-4)^2+(6-7)^2} = \sqrt{2^2+(-1)^2} = \sqrt{4+1} = \sqrt{5}\).

Check if the triangle is right-angled

If the triangle is right-angled, the Pythagorean theorem should hold: \(a^2+b^2=c^2\), where c is the length of the hypotenuse. By comparing, we have: \(AB^2=5^2=25\), \(BC^2=20\), and \(AC^2=5\). Let's test the combinations: \(AB^2+AC^2 = 25+5 = 30 \neq BC^2\). \(AB^2+BC^2=25+20=45 \neq AC^2\). \(BC^2+AC^2 = 20+5=25 \neq AB^2\). So, the triangle is not right-angled.

Check if the angle is obtuse or acute

If $a^2+b^2Nohnone of}=

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly essential when dealing with right-angled triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, if 'c' represents the length of the hypotenuse and 'a' and 'b' are the lengths of the other two sides, the theorem is expressed as:
\[ c^2 = a^2 + b^2 \]
In terms of practical application, this theorem helps us to determine whether a triangle is right-angled by checking this relationship between its sides. Should you come upon any three points that form a triangle, you can use their coordinates to calculate the lengths of the sides and apply the Pythagorean theorem to ascertain if it's a right-angled triangle. This approach is an excellent example of the theorem's utility in coordinate geometry.

Distance Formula

The distance formula is a powerful tool derived from the Pythagorean theorem and is used in coordinate geometry to find the exact distance between two points on a Cartesian plane. It creates a right-angled triangle, with the line segment between the two points as the hypotenuse, to calculate the distance. The formula is as follows:
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
Here, \(x_1, y_1)\) and \(x_2, y_2)\) are the coordinates of the two points. When solving geometry problems, such as identifying the type of triangle made by three points, the distance formula is essential. It helps you determine the lengths of the sides of the triangle and thus forms the basis for further classification using the Pythagorean theorem or other analytical methods.

Geometry

Geometry is an area of mathematics that focuses on the properties and relations of points, lines, surfaces, solids, and higher-dimensional analogs. When classifying triangles, geometry provides a set of criteria based on angles and sides. Triangles are primarily classified into three categories based on their angle measurements:

• Right-angled triangle, where one angle is exactly 90 degrees.
• Acute-angled triangle, where all angles are less than 90 degrees.
• Obtuse-angled triangle, where one angle is greater than 90 degrees.

To determine a triangle's type in a coordinate plane, we often use the concepts of the Pythagorean theorem and the distance formula to calculate side lengths and then analyze these lengths to infer the triangle's internal angle sizes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This mathematical concept blends algebra and geometry, allowing for the solving of geometric problems through the use of algebraic equations. In the context of triangles, coordinate geometry enables us to calculate the lengths of sides and determine angles using algebraic methods. With the distance formula, we can find side lengths; then, by applying the Pythagorean theorem and other properties, we can classify triangles into various types. It's a powerful tool for working with geometric figures when their vertices are given as coordinates on a plane.
Addressing the types of triangles from the lens of coordinate geometry provides a methodical approach that's applicable in many fields, including computer graphics, geographic mapping, and CAD design, where geometric figures are often defined in a coordinate space.