Explanation:
Because PA and PB are tangent to circle O, angles PAO and PBO must be right angles; therefore, triangles PAO and PBO are both right triangles.
Since AO and OB are both radii of circle O, they are congruent. Furthermore, because PA and PB are external tangents originating from the same point, they must also be congruent.
So, in triangles PAO and PBO, we have two sides that are congruent, and we have a congruent angle (all right angles are congruent) between them. Therefore, by the Side-Angle-Side (SAS) Theorem of congruency, triangles PAO and PBO are congruent.
Notice that quadrilateral PAOB can be broken up into triangles PAO and PBO. Since those triangles are congruent, each must comprise one half of the area of quadrilateral PAOB. As a result, if we find the area of one of the triangles, we can double it in order to find the area of the quadrilateral.
Let's determine the area of triangle PAO. We have already established that it is a right triangle. We are told that PO, which is the hypotenuse of the triangle, is equal to 17. We are also told that the diameter of circle O is 16, which means that every radius of the circle is 8, because a radius is half the size of a diameter. Since segment AO is a radius, its length must be 8.
So, triangle PAO is a right triangle with a hypotenuse of 17 and a leg of 8. We can use the Pythagorean Theorem in order to find the other leg. According to the Pythagorean Theorem, if a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse, then:
a2 + b2 = c2
Let us let b represent the length of PA.
82 + b2 = 172
64 + b2 = 289
Subtract 64 from both sides.
b2 = 225
Take the square root of both sides.
b = 15
This means that the length of PA is 15.
Now let's apply the formula for the area of a right triangle. Because the legs of a right triangle are perpendicular, one can be considered the base, and the other can be considered the height of the triangle.
area of triangle PAO = (1/2)bh
= (1/2)(8)(15) = 60
Ultimately, we must find the area of quadrilateral PAOB; however, we previously determined that triangles PAO and PBO each comprise half of the quadrilateral. Thus, if we double the area of PAO, we would get the area of quadrilateral PAOB.
Area of PAOB = 2(area of PAO)
= 2(60) = 120 square units
The answer is 120.
Page 2
The length of one leg of an equilateral triangle is 6. What is the area of the triangle?
Possible Answers:
Correct answer:
Explanation:
The base is equal to 6.
The height of an quilateral triangle is equal to
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The Pythagoras theorem which is also referred to as the Pythagorean theorem explains the relationship between the three sides of a right-angled triangle. According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle. Let us learn more about the Pythagoras theorem, its derivations, and equations followed by solved examples on the Pythagoras theorem triangle and squares.
What is the Pythagoras Theorem?
The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. Observe the following triangle ABC, in which we have BC2 = AB2 + AC2. Here, AB is the base, AC is the altitude (height), and BC is the hypotenuse. It is to be noted that the hypotenuse is the longest side of a right-angled triangle.
Pythagoras Theorem Equation
The Pythagoras theorem equation is expressed as, c2 = a2 + b2, where 'c' = hypotenuse of the right triangle and 'a' and 'b' are the other two legs. Hence, any triangle with one angle equal to 90 degrees produces a Pythagoras triangle and the Pythagoras equation can be applied in the triangle.
History of Pythagoras Theorem
Pythagoras theorem was introduced by the Greek Mathematician Pythagoras of Samos. He was an ancient Ionian Greek philosopher. He formed a group of mathematicians who works religiously on numbers and lived like monks. Finally, the Greek Mathematician stated the theorem hence it was named after him as the "Pythagoras theorem." Though it was introduced many centuries ago its application in the current era is obligatory to deal with pragmatic situations.
Although Pythagoras introduced and popularised the theorem, there is sufficient evidence proving its existence in other civilizations, 1000 years before Pythagoras was born. The oldest known evidence dates back to between 20th to 16th Century B.C in the Old Babylonian Period.
Pythagorean Theorem Formula
The Pythagoras theorem formula states that in a right triangle ABC, the square of the hypotenuse is equal to the sum of the square of the other two legs. If AB and AC are the sides and BC is the hypotenuse of the triangle, then: BC2 = AB2 + AC2. In this case, AB is the base, AC is the altitude or the height, and BC is the hypotenuse.
Another way to understand the Pythagorean theorem formula is using the following figure which shows that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides of the right triangle.
In a right-angled triangle, the Pythagoras Theorem Formula is expressed as:
c2 = a2 + b2
Where,
- 'c' = hypotenuse of the right triangle
- 'a' and 'b' are the other two legs.
Pythagoras Theorem Proof
Pythagoras theorem can be proved in many ways. Some of the most common and widely used methods are the algebraic method and the similar triangles method. Let us have a look at both these methods individually in order to understand the proof of this theorem.
Proof of Pythagorean Theorem Formula using the Algebraic Method
The proof of the Pythagoras theorem can be derived using the algebraic method. For example, let us use the values a, b, and c as shown in the following figure and follow the steps given below:
- Step 1: Arrange four congruent right triangles in the given square PQRS, whose side is a + b. The four right triangles have 'b' as the base, 'a' as the height and, 'c' as the hypotenuse.
- Step 2: The 4 triangles form the inner square WXYZ as shown, with 'c' as the four sides.
- Step 3: The area of the square WXYZ by arranging the four triangles is c2.
- Step 4: The area of the square PQRS with side (a + b) = Area of 4 triangles + Area of the square WXYZ with side 'c'. This means (a + b)2 = [4 × 1/2 × (a × b)] + c2.This leads to a2 + b2 + 2ab = 2ab + c2. Therefore, a2 + b2 = c2. Hence proved.
Pythagorean Theorem Formula Proof using Similar Triangles
Two triangles are said to be similar if their corresponding angles are of equal measure and their corresponding sides are in the same ratio. Also, if the angles are of the same measure, then by using the sine law, we can say that the corresponding sides will also be in the same ratio. Hence, corresponding angles in similar triangles lead us to equal ratios of side lengths.
Derivation of Pythagorean Theorem Formula
Consider a right-angled triangle ABC, right-angled at B. Draw a perpendicular BD meeting AC at D.
In △ABD and △ACB,
- ∠A = ∠A (common)
- ∠ADB = ∠ABC (both are right angles)
Thus, △ABD ∼ △ACB (by AA similarity criterion)
Similarly, we can prove △BCD ∼ △ACB.
Thus △ABD ∼ △ACB, Therefore, AD/AB = AB/AC. We can say that AD × AC = AB2.
Similarly, △BCD ∼ △ACB. Therefore,CD/BC = BC/AC. We can also say that CD × AC = BC2.
Adding these 2 equations, we get AB2 + BC2 = (AD × AC) + (CD × AC)
AB2 + BC2 =AC(AD +DC)
AB2 + BC2 =AC2
Hence proved.
Pythagoras Theorem Triangles
Right triangles follow the rule of the Pythagoras theorem and they are called Pythagoras theorem triangles. The three sides of such a triangle are collectively called Pythagoras triples. All the Pythagoras theorem triangles follow the Pythagoras theorem which says that the square of the hypotenuse is equal to the sum of the two sides of the right-angled triangle. This can be expressed as c2 = a2 + b2; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the triangle.
Pythagoras Theorem Squares
As per the Pythagorean theorem, the area of the square which is built upon the hypotenuse of a right triangle is equal to the sum of the area of the squares built upon the other two sides. These squares are known as Pythagoras squares.
Applications of Pythagoras Theorem
The applications of the Pythagoras theorem can be seen in our day-to-day life. Here are some of the applications of the Pythagoras theorem.
- Engineering and Construction fields
Most architects use the technique of the Pythagorean theorem to find the unknown dimensions. When length or breadth are known it is very easy to calculate the diameter of a particular sector. It is mainly used in two dimensions in engineering fields.
- Face recognition in security cameras
The face recognition feature in security cameras uses the concept of the Pythagorean theorem, that is, the distance between the security camera and the location of the person is noted and well-projected through the lens using the concept.
- Woodwork and interior designing
The Pythagoras concept is applied in interior designing and the architecture of houses and buildings.
People traveling in the sea use this technique to find the shortest distance and route to proceed to their concerned places.
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Example 1: The hypotenuse of a right-angled triangle is 16 units and one of the sides of the triangle is 8 units. Find the measure of the third side using the Pythagoras theorem formula.
Solution:
Given : Hypotenuse = 16 units Let us consider the given side of a triangle as the perpendicular height = 8 units On substituting the given dimensions to the Pythagoras theorem formula
Hypotenuse2 = Base2 + Height2
162 = B2 + 82
B2 = 256 - 64 B = √192 = 13.856 unitsTherefore, the measure of the third side of a triangle is 13.856 units.
Example 2: Julie wanted to wash her building window which is 12 feet off the ground. She has a ladder that is 13 feet long. How far should she place the base of the ladder away from the building?
Solution:
We can visualize this scenario as a right triangle. We need to find the base of the right triangle formed. We know that, Hypotenuse2 = Base2 + Height2. Thus, we can say that b2 = 132 - 122 where 'b' is the distance of the base of the ladder from the feet of the wall of the building. So, b2 = 132 - 122 can be solved as, b2 = 169 - 144 = 25. This means, b = √25 = 5. Hence, we get 'b' = 5.
Therefore, the base of the ladder is 5 feet away from the building.
Example 3: Use the Pythagoras theorem to find the hypotenuse of the triangle in which the sides are 8 units and 6 units respectively.
Solution:
Using the Pythagoras theorem, Hypotenuse2 = Base2 + Height2 = 82 + 62. This leads to Hypotenuse2 = 64 + 36 = 100. Therefore, hypotenuse = √100 = 10 units.
Therefore, the length of the hypotenuse is 10 units.
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FAQs on Pythagoras Theorem
The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be expressed as, c2 = a2 + b2; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the triangle. These triangles are also known as Pythagoras theorem triangles.
What is the Converse of Pythagoras Theorem?
The converse of Pythagoras theorem is: If the sum of the squares of any two sides of a triangle is equal to the square to the third (largest) side, then it is said to be a right-angled triangle.
What is the Use of the Pythagorean Theorem Formula?
The Pythagoras theorem works only for right-angled triangles. When any two values are known, we can apply the Pythagoras theorem and calculate the unknown sides of the triangle. There are other real-life applications of the Pythagoras theorem like in the field of engineering and architecture.
What are the Applications of the Pythagorean Theorem in Real Life?
The Pythagorean theorem is used in various fields. A few of its applications are given below.
- Architecture, construction and Navigation industries.
- For computing the distance between points on the plane.
- For calculating the perimeter, the surface area, the volume of geometrical shapes, and so on.
Can the Pythagorean Theorem Formula be Applied to any Triangle?
No, the Pythagorean theorem can only be applied to a right-angled triangle since the Pythagorean theorem expresses the relationship between the sides of the triangle where the square of the two legs is equal to the square of the third side which is the hypotenuse.
How to Work Out Pythagoras Theorem?
Pythagoras theorem can be used to find the unknown side of a right-angled triangle. For example, if two legs of a right-angled triangle are given as 4 units and 6 units, then the hypotenuse (the third side) can be calculated using the formula, c2 = a2 + b2; where 'c' is the hypotenuse and 'a' and 'b' are the two legs. Substituting the values in the formula, c2 = a2 + b2 = c2 = 42 + 62 = 16 + 36 = √52 = 7.2 units.
What is the Formula of Pythagoras Theorem?
The formula of Pythagoras theorem is expressed as, Hypotenuse2 = Base2 + Height2. This is also written as, c2 = a2 + b2; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the right-angled triangle. Using the Pythagoras theorem formula, any unknown side of a right-angled can be calculated if the other two sides are given.
Why is the Pythagoras Theorem Important?
The Pythagoras theorem is important because it helps in calculating the unknown side of a right-angled triangle. It has other real-life applications in the field of architecture and engineering, navigation, and so on.