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Two triangles are similar if they have:
- all their angles equal
- corresponding sides are in the same ratio
But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough.
There are three ways to find if two triangles are similar: AA, SAS and SSS:
AA
AA stands for "angle, angle" and means that the triangles have two of their angles equal.
If two triangles have two of their angles equal, the triangles are similar.
So AA could also be called AAA (because when two angles are equal, all three angles must be equal).
SAS
SAS stands for "side, angle, side" and means that we have two triangles where:
- the ratio between two sides is the same as the ratio between another two sides
- and we we also know the included angles are equal.
If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
In this example we can see that:
- one pair of sides is in the ratio of 21 : 14 = 3 : 2
- another pair of sides is in the ratio of 15 : 10 = 3 : 2
- there is a matching angle of 75° in between them
So there is enough information to tell us that the two triangles are similar.
Using Trigonometry
We could also use Trigonometry to calculate the other two sides using the Law of Cosines:
In Triangle ABC:
- a2 = b2 + c2 - 2bc cos A
- a2 = 212 + 152 - 2 × 21 × 15 × Cos75°
- a2 = 441 + 225 - 630 × 0.2588...
- a2 = 666 - 163.055...
- a2 = 502.944...
- So a = √502.94 = 22.426...
In Triangle XYZ:
- x2 = y2 + z2 - 2yz cos X
- x2 = 142 + 102 - 2 × 14 × 10 × Cos75°
- x2 = 196 + 100 - 280 × 0.2588...
- x2 = 296 - 72.469...
- x2 = 223.530...
- So x = √223.530... = 14.950...
Now let us check the ratio of those two sides:
a : x = 22.426... : 14.950... = 3 : 2
the same ratio as before!
Note: we can also use the Law of Sines to show that the other two angles are equal.
SSS
SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
In this example, the ratios of sides are:
- a : x = 6 : 7.5 = 12 : 15 = 4 : 5
- b : y = 8 : 10 = 4 : 5
- c : z = 4 : 5
These ratios are all equal, so the two triangles are similar.
Using Trigonometry
Using Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:
In Triangle ABC:
- cos A = (b2 + c2 - a2)/2bc
- cos A = (82 + 42 - 62)/(2× 8 × 4)
- cos A = (64 + 16 - 36)/64
- cos A = 44/64
- cos A = 0.6875
- So Angle A = 46.6°
In Triangle XYZ:
- cos X = (y2 + z2 - x2)/2yz
- cos X = (102 + 52 - 7.52)/(2× 10 × 5)
- cos X = (100 + 25 - 56.25)/100
- cos X = 68.75/100
- cos X = 0.6875
- So Angle X = 46.6°
So angles A and X are equal!
Similarly we can show that angles B and Y are equal, and angles C and Z are equal.
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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 Formula
In 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:
- The area of a right triangle with base 6 ft and height 4 ft is 1/2 × 6 × 4 = 12 ft2.
- The area of a right triangle with base 10 m and height 5 m is 1/2 × 10 × 5 = 25 m2.
- The area of a right triangle with base 11 in and height 5 in is 1/2 × 11 × 5 = 27.5 in2.
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 Hypotenuse
Let 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.
-
Example 1: The longest side of a bread slice that resembles a right triangle is 13 units. If its height is 12 units, find its area using the area of a right triangle formula.
Solution:
We know that the longest side of a right triangle is called the hypotenuse.
So, it is given that hypotenuse = 13 units and height = 12 units.
Substitute the given values in the Pythagoras theorem,
(hypotenuse)2 = (base)2 + (height)2
132 = (base)2 + (12)2
169 = (base)2 + 144
(base)2 = 25
base = √(25) = 5 units.
The area of the bread slice = 1/2 × base × height = 1/2 × 5 × 12 = 30 square units.
Therefore, the area of the given bread slice = 30 square units.
-
Example 2: A swimming pool is in the shape of a right triangle. Its sides are in the ratio 3:4:5. Its perimeter is 720 units. Find its area.
Solution:
Let us assume that the sides of the swimming pool be 3x, 4x, and 5x.
It is given that its perimeter = 720 units.
3x + 4x + 5x = 720
12x = 720
x = 60
So the sides of the triangle are,
3x = 3(60) = 180 units
4x = 4(60) = 240 units
5x = 5(60) = 300 units
Since 300 units is the longest side of the swimming pool (which is in the shape of a right triangle), it is the hypotenuse.
So, 180 units and 240 units must be the base and the height of the swimming pool interchangeably.
Using the area of right triangle formula,
The area of the swimming pool = 1/2 × base × height = 1/2 × 180 × 240 = 21,600 units2.
Therefore, the area of the given swimming pool = 21,600 units2.
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FAQs on Area of Right Triangle
The 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.
- If we know the base and the hypotenuse, we find the height using the Pythagoras theorem.
- If we know the height and the hypotenuse, we find the base using the Pythagoras theorem.
Then, we can find the area of the right triangle using the formula 1/2 × base × height.