Tag: Right Angle

Introduction

A straight line that intersects another straight line at a 90-degree angle is said to be perpendicular to the first line. The small square in the middle of two perpendicular lines in the figure represents 90 degrees, also known as a right angle. Here, two lines cross at a right angle, indicating that they are perpendicular to one another.

In contrast to sloping or horizontal lines or surfaces, perpendicular lines or surfaces point directly upward. An object is at a 90-degree angle to another if it is perpendicular to it. A pair of lines, vectors, planes, or other objects are said to be perpendicular if they intersect at a right angle. Two vectors are perpendicular if their dot product equals zero.

Perpendiculars

When two lines intersect at a right angle, they are said to be perpendicular to one another. A first line is perpendicular to a second line, more specifically, if the two lines intersect and the straight angle on one side of the first line is split into two congruent angles by the second line at the intersection. Since perpendicularity is symmetric, if one line is perpendicular to another, the other line is also perpendicular to the first. As a result, we don’t need to specify an order when referring to two lines as perpendicular (to one another).

Perpendicular Lines

Segments and rays are easily extended by perpendicularity. If a line segment AB and a line segment CD result in an infinite line when both directions are extended, then the two resulting lines are perpendicular in the sense mentioned above. Line segment AB is perpendicular to line segment CD and can be represented by the symbol AB ⊥ CD. If a line crosses every other line in a plane, it is said to be perpendicular to the plane. The definition of line perpendicularity is necessary for understanding this definition.

Perpendicular Theorem

According to the perpendicular line theorem, two straight lines are perpendicular to one another if they intersect at a point and create a pair of equal linear angles.

Assume two lines AB and CD intersect each other at O, such that ∠AOC = ∠COB, also since AB is a line, ∠AOC and ∠COB also form a linear pair.

Then, ∠AOC + ∠COB = 180°

Using ∠COB = ∠AOC

⇒ ∠AOC + ∠AOC = 180°

⇒ 2 ∠AOC = 180°

⇒ ∠AOC = 90°

Thus, since the angle of intersection is 90°, we can say that AB is perpendicular to CD and vice versa.

Also Read: Related Angles

Perpendicularity: Slope Formula

Perpendicularity is known as the mathematical condition that two lines need to satisfy to be called perpendicular. Mathematically, if two lines are perpendicular to each other, then the product of their slopes is negative unity.

For example, let two lines of slope . Then these lines are said to be perpendiculars to one another if their slopes have a product -1, i.e.,  

Equation of a Perpendicular Line.

Using the conditions from previous sections, we can find the equation of the perpendicular line to any given line’s equation, at a certain point.

Let, ax + by = c be a line, and we need to find a line perpendicular to it passing through

First, we will find the slope of the given line, 

Slope of a line m = -a/b

Now, using perpendicularity, if the slope of the second line is m’, then for these lines to be perpendicular

m × m’=- 1

⇒ m’ =- 1/m =- 1/-a/b = b/a

Thus, the slope of the perpendicular line is, 

m’= b/a

Then, we have a point as well as the slope for the equation of the perpendicular line,

Using point-slope form

If we know the exact values of a, b and then we can further simplify this equation.

Interesting Facts about Perpendicular Lines

• In order to obtain the maximum support for the roof, walls and pillars are constructed perpendicular to the ground in our homes and other buildings. This is just one example of how perpendiculars are used in everyday life.
• Perpendiculars of two lines that meet at an angle will also meet at that same angle.

Solved Examples

Example: Which of the following pair of lines are perpendicular, parallel, or simply intersecting?

1. Intersecting, since the angle of intersection here is given to be 100 degrees.
2. Perpendicular, as we know the right angle is also represented by a small square, we can say that these lines are perpendicular to each other.
3. Parallel lines, clearly extending these lines to infinity we will never see them intersecting; thus, they are parallel.
4. Perpendicular, clearly the angle of intersection here is given to be 90 degrees.

Summary

The subject of perpendiculars and the perpendicularity of lines were covered in this article. The reader should be able to comprehend the meaning of perpendicular lines, the symbol used to represent them, as well as the formula and theorems relating to perpendicular lines, after carefully reading this article. Two lines are said to be perpendicular if their angle of intersection is a right angle. The slopes of perpendicular lines are negative reciprocals of each other.

Frequently Asked Questions (FAQs)

1.What are Perpendicular Lines?

Ans. When two lines intersect at an angle of 90 degrees, the lines are said to be perpendicular to each other.

2. How do you Find the Slope of a line Perpendicular to a Given Line?

Ans. The slope of perpendicular lines are negative reciprocals of each other; thus, the slope of a perpendicular line can be found simply by negating the reciprocal of the slope of the given line.

3. What are Perpendicular Bisectors?

Ans. A line that divides another line segment into two halves while also being at a right angle to it is known as the perpendicular bisector of the line segment.

4. Are all Intersecting Lines Perpendicular?

Ans. No, all intersecting lines are not perpendicular, but all perpendicular lines are intersecting, that too at a specific angle, i.e., 90 degree.

Also read: Properties, Area of Right-Angled Triangles

Introduction

Pythagoras, a Greek philosopher who was born around 570 BC, is remembered by the theorem’s name. The theorem has likely been proved the most times of any mathematical theorem using a variety of techniques. The proofs are numerous, some of which go back thousands of years, and include both geometric and algebraic proofs. The Pythagorean theorem is extremely useful when determining the shortest distance between two points or the degree of the mountain slope. In a right-angle triangle the square of the hypotenuse is said to be equal to the sum of the squares of the two legs.

Right Angle Triangle

A triangle with a right angle is one in which one angle is 90 degrees. We refer to this triangle as a right-angle triangle since 90 is also referred to as the right angle. Triangle sides with a right angle were given unique names. The side directly opposite the right angle is known as the hypotenuse. Based on the values of the various sides, the right triangles are divided into isosceles and scalene types.

Properties of Right-Angle Triangle

• The height, base, and hypotenuse of a right-angle triangle are its three sides. 
• The two adjacent sides are referred to as base and height or perpendicular.
• Three similar right triangles are formed if we draw a perpendicular line from the vertex of a right angle to the hypotenuse.
• The radius of a circle whose circumference includes all three vertices is equal to one-half the length of the hypotenuse.
• The triangle is known as an isosceles right-angled triangle, where the adjacent sides to the 90° are equal in length if one of the angles is 90° and the other two angles are each equal to 45°.

Pythagoras Theorem

Pythagoras is a potent theorem that establishes the relationship between the sides of a right-angle triangle. According to Pythagoras’ theorem –

“Square of the hypotenuse is equal to the sum of the square of the other two legs of the right angle triangle”. Mathematically, it may be expressed as

              Hypotenuse² = Perpendicular² + Base² 

Area of the Right-Angle Triangle

The area of the right-angle triangle is the region enclosed within the triangle’s perimeter. The formula for a right-angle triangle’s area is.

Area of right-angle triangle = (Base × Perpendicular)

Facts

• A triangle must be a right triangle if it obeys Pythagoras’ theorem.
• The longest side of a triangle is the one that makes the largest angle.
• When the midpoint of the hypotenuse of a right-angled triangle is joined to the vertex of the right angle, the resulting line segment is half of the hypotenuse. In other words, the center of the hypotenuse is the circumcenter of the right-angled triangle.
• If two sides of a right angle are known, we can find the other side using Pythagoras’ Theorem.
• From the provided value of sides, we may determine whether a right-angle triangle is possible.

Summary

A right-angled triangle is one in which one of the angles is a right angle (90 degrees), and the hypotenuse is the side opposite to the right angle. The hypotenuse square of a right-angled triangle is equal to the sum of the squares of the other two sides, according to Pythagoras’ Theorem.

Solved Examples

Example 1: In the right-angle triangle, If PQ = 5 cm and QR = 12 cm, then what is the value of PR?

Solution:  By Pythagoras theorem, we have, 

   Hypotenuse² = Perpendicular² + Base²

PR² = 5² + 12²

PR² = 25 + 144

PR =  = 13 cm

Hence, the value of PR is 13 cm.

Example 2: If a triangle has three sides 9cm, 5 cm, and 7 am respectively, check whether the triangle is a right triangle or not.

Solution: According to the theorem, if the square of the longest side equals the sum of the squares of the other two sides, a triangle is said to be, a right triangle. 

9² = 5² + 7² 

81 = 25 + 49

81 ≠ 74

 Thus, 81 is not equal to 74. Hence, the given triangle is not a right-angle triangle.

Frequently Asked Questions

1.Which Side of a Right-Angled Triangle is the Longest?

Ans: The hypotenuse of a right-angled triangle is its longest side.

2.What is a Right-Angled Triangle’s Perimeter?

Ans: The perimeter of a triangle is the sum of all sides.

Perimeter = base + perpendicular + hypotenuse.

3.Can there be two Right Angles in a Triangle? Explain.

Ans: No, there can never be two right angles in a triangle. A triangle has exactly three sides and interior angles that add up to 180 degrees. This means that if a triangle contains two right angles, the third angle must be zero degrees, which means that the third side will overlap the opposite side. Therefore, a triangle with two right angles is not possible.