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Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... geometry/images/pyth1.js ... then the biggest square has the exact same area as the other two squares put together! It is called "Pythagoras' Theorem" and can be written in one short equation: a2 + b2 = c2 Note:
DefinitionThe longest side of the triangle is called the "hypotenuse", so the formal definition is:
In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. Sure ... ?Let's see if it really works using an example.
Why Is This Useful?If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!) How Do I Use it?Write it down as an equation:
Start with:a2 + b2 = c2 Put in what we know:52 + 122 = c2 Calculate squares:25 + 144 = c2 25+144=169:169 = c2 Swap sides:c2 = 169 Square root of both sides:c = √169 Calculate:c = 13 Read Builder's Mathematics to see practical uses for this. Also read about Squares and Square Roots to find out why √169 = 13
Start with:a2 + b2 = c2 Put in what we know:92 + b2 = 152 Calculate squares:81 + b2 = 225 Take 81 from both sides: 81 − 81 + b2 = 225 − 81 Calculate: b2 = 144 Square root of both sides:b = √144 Calculate:b = 12
Start with:a2 + b2 = c2 Put in what we know:12 + 12 = c2 Calculate squares:1 + 1 = c2 1+1=2: 2 = c2 Swap sides: c2 = 2 Square root of both sides:c = √2 Which is about:c = 1.4142... It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.
Does a2 + b2 = c2 ?
They are equal, so ... Yes, it does have a Right Angle!
Does 82 + 152 = 162 ?
So, NO, it does not have a Right Angle
Does a2 + b2 = c2 ? Does (√3)2 + (√5)2 = (√8)2 ? Does 3 + 5 = 8 ? Yes, it does! So this is a right-angled triangle Get paper pen and scissors, then using the following animation as a guide:
Another, Amazingly Simple, ProofHere is one of the oldest proofs that the square on the long side has the same area as the other squares. Watch the animation, and pay attention when the triangles start sliding around. You may want to watch the animation a few times to understand what is happening. The purple triangle is the important one.
We also have a proof by adding up the areas. Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived. 511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361 Activity: Pythagoras' Theorem Copyright © 2022 Rod Pierce
The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs of the right triangle. This same relationship is often used in the construction industry and is referred to as the 3-4-5 Rule.
The right triangle below has one leg with a length of three, another leg with a length of four and a hypotenuse with a length of five. Given the lengths of any two sides of a right triangle, the length of the third side can be calculated using the Pythagorean theorem. In the example above, there are three possible unknowns. Each case is outlined below. There are many ways to prove the Pythagorean theorem. One such proof is given here.
The area of a right triangle is the portion that is covered inside the boundary of the triangle. A right-angled triangle is a triangle where one of the angles is a right angle (90 degrees). It is simply known as a right triangle. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called legs. The two legs can be interchangeably called base and height. The area of right-angle triangle formula is given in the image below. What is Area of a Right Triangle?The area of a right-angled triangle, as we discussed earlier, is the space that is inside it. This space is divided into squares of unit length and the number of unit squares that are inside the right triangle is its area. The area is measured in square units. Let us consider the following right triangle whose base is 4 units and height is 3 units. Can you try counting the number of unit squares inside this triangle? There are 6 unit squares in total. So the area of the above triangle is 6 square units. But it is not possible to calculate the area of a right triangle always by counting the number of squares. There must be a formula to do this. Let us see what is the formula for finding the area of a right triangle. Area of Right Triangle FormulaIn the above example, if we multiply the base and height, we get 3 × 4 = 12 and if we divide it by 2, we get 6. So the area of a right triangle is obtained by multiplying its base and height and then making the product half. Area of a right triangle = 1/2 × base × height Examples:
How to Derive Area of Right Triangle Formula?Consider a rectangle of length l and width w. Also, draw a diagonal. You can see that the rectangle is divided into two right triangles. We know that the area of a rectangle is length × width. So the area of the above rectangle is l × w. We can see that the two right triangles are congruent as they can be arranged such that one overlaps the other. Thus, the area of the rectangle is equal to twice the area of one of the above right triangles. i.e., Area of rectangle = l × w = 2 × (Area of one right triangle) This gives, Area of one right triangle = 1/2 × l × w. We usually represent the legs of the right-angled triangle as base and height. Thus, the formula for the area of a right triangle is, Area of a right triangle = 1/2 × base × height. Area of Right Triangle With HypotenuseLet us recollect the Pythagoras theorem which states that in a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides. i.e., (hypotenuse)2 = (base)2 + (height)2. Though it is not possible to find the area of a right triangle just with the hypotenuse, it is possible to find its area if we know one of the base and height along with the hypotenuse. Let us see an example. Example: Find the area of a right angle triangle whose base is 6 in and hypotenuse is 10 in. Solution: Substitute the given values in the Pythagoras theorem, (hypotenuse)2 = (base)2 + (height)2 102 = 62 + (height)2 100 = 36 + (height)2 (height)2 = 64 height = √(64) = 8 in. So, the area of the given triangle = 1/2 × base × height = 1/2 × 6 × 8 = 24 in2.
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FAQs on Area of Right TriangleThe area of a right triangle is defined as the total space or region covered by a right-angled triangle. It is expressed in square units. Some common units used to represent area are m2, cm2, in2, yd2, etc. What is the Formula for Finding the Area of a Right Triangle?The area of a right triangle of base b and height h is 1/2 × base × height (or) 1/2 × b × h square units. How Do You Find the Perimeter and Area of a Right Triangle?The area of a right triangle of base b and height h is found using the formula 1/2 × b × h and its perimeter is obtained by just adding all the sides. In case only two of its sides are given, then we use the Pythagoras theorem to find the third side. How Do You Find the Area of a Right Triangle Without the Base?If only the height and hypotenuse of a right triangle are given, then before finding the area of the triangle, we first need to find the base using the Pythagoras theorem. Then we can use the formula 1/2 × base × height to find its area. For example, to find the area of a right triangle with a height of 4 cm and hypotenuse 5 cm, we first find its base using the Pythagoras theorem. Then we get, base = √[(hypotenuse)2 - (height)2] = √(52 - 42) = √9 = 3 cm. Area of the right triangle = 1/2 × 3 × 4 = 6 cm2. How Do You Find the Area of a Right Triangle Without the Height?If only the base and hypotenuse of a right triangle are given, then before finding the area of the triangle, we first need to find the height using the Pythagoras theorem. Then we can use the formula 1/2 × base x height to find its area. For example, to find the area of a right triangle with a base of 4 cm and hypotenuse 5 cm, we first find its height using the Pythagoras theorem. Then we get height = √[(hypotenuse)2 - (base)2] = √(52 - 42) = √9 = 3 cm. Area of the triangle = 1/2 × 3 × 4 = 6 cm2. How Do You Find the Area of a Right Triangle With a Hypotenuse?In fact, it is not possible to find the area of a right triangle just with the hypotenuse. We need to know at least one of the base and height along with the hypotenuse to find the area.
Then, we can find the area of the right triangle using the formula 1/2 × base × height. |
What is the square of a right triangle?
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