Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A triangle with vertices \((4,0),(-1,-1),(3,5)\) is: (A) isosceles and right angled (B) isosceles but not right angled (C) right angled but not isosceles (D) neither right angled nor isosceles

Step-by-Step Solution

Answer

The triangle is isosceles and right angled. (Option A)

Step 1: Calculate Length of Sides

First, we find the side lengths of triangle by using the distance formula: \[AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Using this formula, we determine:For side \(AB\) with points \((4,0)\) and \((-1,-1)\):\[AB = \sqrt{(4+1)^2 + (0+1)^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}\]For side \(BC\) with points \((-1,-1)\) and \((3,5)\):\[BC = \sqrt{(3+1)^2 + (5+1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}\]For side \(CA\) with points \((3,5)\) and \((4,0)\):\[CA = \sqrt{(4-3)^2 + (0-5)^2} = \sqrt{1^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26}\]

Step 2: Determine If the Triangle is Isosceles

The triangle is isosceles if any two sides are equal in length. From our calculations:\[AB = \sqrt{26}, BC = \sqrt{52}, CA = \sqrt{26}\]Since \(AB = CA\), the triangle is isosceles.

Step 3: Check for Right Angle

A triangle is right-angled if the Pythagorean theorem holds for its sides. We check using:\[AB^2 + CA^2 = BC^2\]\[ (\sqrt{26})^2 + (\sqrt{26})^2 = (\sqrt{52})^2\]\[26 + 26 = 52\]Since this equation holds, the triangle is also right-angled.

Key Concepts

Distance Formula

To determine the length of a line segment between two points on a coordinate plane, the distance formula is used. This formula is derived from the Pythagorean theorem and helps find the distance by measuring the hypotenuse of a right-angled triangle formed by the difference in the points' x and y coordinates. The formula is expressed as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \( d \) represents the distance between points \((x_1, y_1)\) and \((x_2, y_2)\).

• Identify the coordinates of the two points you want to calculate the distance between.
• Subtract the x-coordinates and square the result.
• Subtract the y-coordinates and square that result too.
• Add those squared results together, then take the square root of the sum to find the distance.

Using this method, you can find the lengths of the sides of a triangle, as seen when calculating sides \(AB\), \(BC\), and \(CA\) in our original exercise.

Isosceles Triangle

An isosceles triangle is a type of triangle in which at least two sides are of equal length. This property also implies that the angles opposite to these equal sides are equal as well. Identifying an isosceles triangle involves:

• Calculating the lengths of the sides using the distance formula.
• Checking if any two side lengths are equal.

In the exercise solution, the triangle was found to be isosceles because the side lengths \(AB\) and \(CA\) were equal, both being \(\sqrt{26}\). Sometimes, observing the symmetry of a triangle or the equality of angles can also indicate that it's isosceles.
Understanding this concept helps in identifying the type of triangle and solving problems related to angles and side lengths.

Right Angled Triangle

A right-angled triangle is one where one of the angles measures exactly 90 degrees. The defining characteristic is that the Pythagorean theorem holds for its sides. The theorem states that for a triangle with sides \(a\), \(b\), and hypotenuse \(c\): \[ a^2 + b^2 = c^2 \] To determine if a triangle is right-angled:

• Identify the longest side; this is usually considered the hypotenuse.
• Square the lengths of all three sides.
• Check if the sum of the squares of the two shorter sides equals the square of the longest side.

In our exercise, the triangle was right-angled because \(AB^2 + CA^2 = BC^2\), which holds true as \(26 + 26 = 52\). Recognizing right-angled triangles is crucial for solving problems involving distances, angles, and sometimes even coordinate geometry.