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Chapter 28: Problem 19

The sides of a triangle are \(3 x+4 y, 4 x+3 y\) and \(5 x+5 y\) where \(x\), \(y>0\) then the triangle is (a) right angled (b) obtuse angled (c) equilateral (d) none of these

Short Answer

Expert verified
The triangle is obtuse-angled.

Step by step solution

01

Understand the Problem

We are given three sides of a triangle: \(3x + 4y\), \(4x + 3y\), and \(5x + 5y\). We need to determine the type of triangle based on these side lengths. We have options: right-angled, obtuse-angled, equilateral, or none of these.
02

Analyze Side Lengths for Equality

An equilateral triangle has all sides equal. Compare the given side lengths:1. Check if \(3x + 4y = 4x + 3y\)2. Check if \(4x + 3y = 5x + 5y\)3. Check if \(3x + 4y = 5x + 5y\)Since none of these equalities hold, the triangle is not equilateral.
03

Check for Right Angle Using Pythagoras Theorem

In a right-angled triangle, the square of the longest side should equal the sum of the squares of the other two. Identify the longest side, which is \(5x + 5y\). Verify:\((5x + 5y)^2 = (3x + 4y)^2 + (4x + 3y)^2\)Calculate both sides and compare:\[(25x^2 + 50xy + 25y^2) eq (9x^2 + 24xy + 16y^2 + 16x^2 + 24xy + 9y^2)\]The equality doesn't hold, so the triangle is not right-angled.
04

Check for Obtuse Angle Using Extended Pythagorean Theorem

For an obtuse-angled triangle, one side squared should be greater than the sum of the squares of the other two sides for the longest side:\((5x + 5y)^2 > (3x + 4y)^2 + (4x + 3y)^2\)Simplifying leads to:\(25x^2 + 50xy + 25y^2 > 25x^2 + 48xy + 25y^2\)This inequality holds, indicating that the triangle is obtuse-angled.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right-Angled Triangle
A right-angled triangle is one that has one angle measuring precisely 90 degrees. This triangle has some fascinating properties:
  • The side opposite the right angle is known as the hypotenuse, and it is the longest side.
  • Pythagoras' Theorem applies: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, this is written as: \(c^2 = a^2 + b^2\).
To determine if a triangle with sides \(3x + 4y, 4x + 3y\) and \(5x + 5y\) is right-angled, we assess whether the longest side follows this rule. However, calculations show that \((5x + 5y)^2\) is not equal to \((3x + 4y)^2 + (4x + 3y)^2\). Consequently, this triangle is not right-angled.
Understanding the structure of right-angled triangles is crucial for solving many geometrical problems, as they simplify calculations using their specific properties.
Obtuse-Angled Triangle
An obtuse-angled triangle has one angle that is greater than 90 degrees. These triangles have some distinct properties:
  • In an obtuse triangle, the side opposite the obtuse angle is the longest.
  • The square of this longest side is greater than the sum of the squares of the other two sides, as per the Extended Pythagorean Theorem. This can be expressed as: \(c^2 > a^2 + b^2\).
In the given triangle with sides \(3x + 4y, 4x + 3y\), and \(5x + 5y\), applying this theorem reveals that \((5x + 5y)^2\) is indeed greater than \((3x + 4y)^2 + (4x + 3y)^2\). Therefore, the triangle is obtuse-angled.
Understanding this concept helps students identify obtuse triangles quickly, especially when they are involved in complex geometric calculations or proofs.
Equilateral Triangle
An equilateral triangle is a triangle where all three sides are of equal length, and each angle measures 60 degrees. This symmetry gives the equilateral triangle unique characteristics:
  • All sides are equal, typically written formulaically as \(a = b = c\).
  • It is perfectly balanced, with each angle being equal, resulting in a high level of symmetry.
For the triangle with sides \(3x + 4y, 4x + 3y\), and \(5x + 5y\), assessing the equality of these sides clearly shows they are not equal, thus ruling out the possibility of it being equilateral.
Recognizing the distinct properties of equilateral triangles is vital for understanding concepts of symmetry and congruence in geometry.

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Most popular questions from this chapter

The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is [2003] (a) \(\frac{a}{4} \cot \left(\frac{\pi}{2 n}\right)\) (b) \(a \cot \left(\frac{\pi}{n}\right)\) (c) \(\frac{a}{2} \cot \left(\frac{\pi}{2 n}\right)\) (d) \(a \cot \left(\frac{\pi}{2 n}\right)\).

A tower \(T_{1}\) of height \(60 \mathrm{~m}\) is located exactly opposite to a tower \(T_{2}\) of height \(80 \mathrm{~m}\) on a straight road. From the top of \(T_{1}\), if the angle of depression of the foot of \(T_{2}\) is twice the angle of elevation of the top of \(T_{2}\), then the width (in \(\mathrm{m}\) ) of the road between the feet of the towers \(T_{1}\) and \(T_{2}\) is \mathrm{\\{} O n l i n e ~ A p r i l ~ 1 5 , ~ 2 0 1 8 ] ~ (a) \(20 \sqrt{2}\) (b) \(10 \sqrt{2}\) (c) \(10 \sqrt{3}\) (d) \(20 \sqrt{3}\)

With the usual notation, in \(\triangle \mathrm{ABC}\), if \(\angle \mathrm{A}+\angle \mathrm{B}=120^{\circ}\) \(\mathrm{a}=\sqrt{3}+1\) and \(\mathrm{b}=\sqrt{3}-1\), then the ratio \(\angle \mathrm{A}: \angle \mathrm{B}\), is: [Jan. 10, 2019 (II)] (a) \(7: 1\) (b) \(5: 3\) (c) \(9: 7\) (d) \(3: 1\)

If the angles of elevation of the top of a tower from three collinear points \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\), on a line leading to the foot of the tower, are \(30^{\circ}, 45^{\circ}\) and \(60^{\circ}\) respectively, then the ratio, \(\mathrm{AB}: \mathrm{BC}\), is: (a) \(1: \sqrt{3}\) (b) \(2: 3\) (c) \(\sqrt{3}: 1\) (d) \(\sqrt{3}: \sqrt{2}\)

The angle of elevation of the top of a vertical tower from a point \(\mathrm{A}\), due east of it is \(45^{\circ}\). The angle of elevation of the top of the same tower from a point \(\mathrm{B}\), due south of \(\mathrm{A}\) is \(30^{\circ}\). If the distance between Aand \(\mathrm{B}\) is \(54 \sqrt{2} \mathrm{~m}\), then the height of the tower (in metres), is: \(\quad\) Online April 10,2016\(]\) (a) 108 (b) \(36 \sqrt{3}\) (c) \(54 \sqrt{3}\) (d) 54 (which are the ex-radii) then
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