Question

Let the vertices of a triangle \(A B C\) be \((4,3),(7,-1),(9,3)\) then the triangle is: (a) Scalene (b) Isosceles (c) Equilateral (d) none of (a), (b),(c)

Short answer

Answer: (b) Isosceles

Step-by-step solution

Calculate the side lengths

Using the distance formula, calculate the lengths of the three sides of the triangle. The distance between two points (x1, y1) and (x2, y2) can be found using the expression: \(distance = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\) Let AB be the distance between points A(4,3) and B(7,-1), AC be the distance between points A(4,3) and C(9,3), and BC be the distance between points B(7,-1) and C(9,3). AB = \(\sqrt{(7 - 4)^2 + (-1 - 3)^2}\) \\ AC = \(\sqrt{(9 - 4)^2 + (3 - 3)^2}\) \\ BC = \(\sqrt{(9 - 7)^2 + (3 + 1)^2}\)

Simplify the side lengths

Simplify the expressions for the side lengths found in Step 1. AB = \(\sqrt{3^2 + (-4)^2}\) = \(\sqrt{9 + 16}\) = \(\sqrt{25}\) = 5 \\ AC = \(\sqrt{5^2 + 0^2}\) = \(\sqrt{25}\) = 5 \\ BC = \(\sqrt{2^2 + 4^2}\) = \(\sqrt{4 + 16}\) = \(\sqrt{20}\) = 2√5

Determine the type of triangle

Based on the side lengths calculated (AB = 5, AC = 5, BC = 2√5), determine the type of triangle formed. Since two sides AB and AC have the same length, and the third side BC has a different length, the triangle is an isosceles triangle. The correct option is (b) Isosceles.

Key concepts

Triangles

In geometry, a triangle is a polygon with three edges and three vertices. It is one of the simplest shapes and is fundamental in mathematics. Triangles can be categorized based on their sides and angles. Depending on the side lengths, triangles can be classified into three types:

• Scalene Triangle: All three sides have different lengths.
• Isosceles Triangle: Two sides are of equal length, while the third side is different.
• Equilateral Triangle: All three sides are the same length.

Triangles are crucial in geometry because they help in understanding the properties of shapes and in solving complex geometric problems. They form the basis for the study of polygons of all kinds. A deep understanding of triangles can enhance comprehension of more complex figures like quadrilaterals, circles, and three-dimensional objects.

Distance Formula

The distance formula is a mathematical tool used to determine the distance between two points in a plane. It is derived from the Pythagorean theorem and is essential for various geometric calculations.
The formula is expressed as:
\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula calculates the straight-line distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a coordinate plane. For instance, it was used to determine the side lengths of triangle \(ABC\) from points with given coordinates.

• Use the formula to calculate each side of the triangle individually.
• Simplify each result to find the exact lengths of these sides.

Understanding and applying the distance formula efficiently can help resolve a wide range of geometric and algebraic problems, including determining triangle types.

Isosceles Triangle

An isosceles triangle is renowned for having two sides of equal length, which consequently means it also has two angles of equal measure. In the exercise above, the triangle \(ABC\) has side lengths AB and AC equal, which confirms its identity as isosceles.
Key features of an isosceles triangle include:

• Two equal sides, known as the "legs."
• The angles opposite these equal sides are equal.
• The third side, called the "base," can vary in length.

Isosceles triangles are not only vital in geometry but also appear in various real-life contexts, like architectural designs and engineering projects. Recognizing and understanding the properties of isosceles triangles helps in solving numerous geometric problems and can simplify further calculations or constructions.