# Right Triangles

When you are on the edge of a rectangular or square lawn and intend on getting to the adjacent end, you instinctively walk diagonally towards the adjacent end because you believe it is the shortest distance. Do you know you form a right triangle when you take this route?

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## What is a right triangle?

A right triangle is a triangle in which one angle is a right angle, that is a 90-degree angle. It is also known as a right-angled triangle.

Right triangles are characterized by a square drawn on the vertex of their right angle as shown below.

An image of a right triangle, StudySmarter Originals

## Types of right triangles

There are two types of right triangles.

### Isosceles right triangle

An isosceles right triangle has two of its sides of equal length. That is, aside from the 90 degrees angle, its interior angles are both 45 degrees each.

An image of an isosceles right triangle - StudySmarter OriginalsIsosceles right triangles are used in finding the sine, cosine, and tangent of the angle of 45 degrees.

### Scalene right triangle

A scalene right triangle has none of its sides equal. This means that one of its interior angles is 90 degrees with the other two not equal but summing up to 90 degrees.

An image of a scalene right triangle, StudySmarter Originals

Scalene right triangles are used in finding the sine, cosine, and tangent of the two special angles 30° and 60°.

## Geometry of right triangles

A right triangle consists of three sides, two complementary angles, and a right angle. The longest side of the triangle is called the hypotenuse, and it is opposite to the right angle within the triangle. The other two sides are referred to as the base and the altitude (or height).

An illustration on the components of a right triangle - StudySmarter Originals

## Properties of right triangles

A triangle can be identified as a right triangle if it verifies the following,

1. One of its angles must be equal to 90 degrees.

2. The non-right angles are acute, that is the measure of each is less than 90 degrees.

Classify the following angles labelled I to III.

1. Right triangles
2. Non-right triangles
3. Isosceles right triangles
4. Scalene right triangles

Solution:

We can see that figure I is a right triangle because it has one of its angles equal to 90°. However, the indications on its sides show that no two of its sides are equal. This means that figure I is a scalene right triangle.

However, in figure II, none of its angles equals 90º. Hence figure II is a non-right triangle.

Likewise what we have in figure I, figure III has one of its angles equal to 90°. This makes it a right triangle. Unlike figure I, figure III has a 45º angle, which means that the third angle would also be 45°. Therefore, this implies that figure III is an isosceles right triangle since it does not just possess one of its angles equal to 90° but the two other angles are equal. Hence the right response to this is question is,

a. Right triangles - I and III

b. Non-right triangle - II

c. Isosceles right triangle - III

d. Scalene right triangle - I

### Perimeter of right triangles

The perimeter of any 2-dimensional surface is the distance around that figure. Thus the perimeter of a right triangle is the sum of all three sides: the height, the base, and the hypotenuse.

So the perimeter for any right triangle with sides a, b, and c is given by

$Perimeter=a+b+c$

A right-angled triangle - StudySmarter Originals

Find the perimeter of the triangle.

Solution:

The perimeter of the triangle is equal to the sum of the lengths of its sides. Thus,

$P=3+4+5=12cm$

### Area of right triangles

The area of a right triangle can be calculated by multiplying the base by the height (or altitude) and dividing the resulting by two.

$A=\frac{Base×Height}{2}.$

In particular, in order to find the area of an isosceles right triangle, you replace either the base with the height or vice versa as the height and the base are of equal length.

A right triangle cement block with sides 5 cm, 13 cm, and 12 cm is used to cover up a square lawn with a side length of 30 cm. How many right triangles are needed to cover the lawn?

Solution:

We need to determine the surface area of the square lawn. We let l be the side length of the square lawn so l = 30m,

${Area}_{squarelawn}={l}^{2}={30}^{2}=900{m}^{2}$

In order to know the number of right triangles that would cover up the square lawn, we should calculate the area of each right triangle that would occupy in order to fill the square.

${Area}_{righttriangle}=\frac{1}{2}×base×height=\frac{1}{2}×12×5=30c{m}^{2}$

Now the area of the right triangle and the square has been calculated, we can now determine how many of the right-triangular cement blocks can be found on the square lawn.

$Numberofcementblock=\frac{Areaofsquarelawn}{Areaofrightangledcementblock}=\frac{Are{a}_{squarelawn}}{Are{a}_{righttriangle}}\phantom{\rule{0ex}{0ex}}$

But first, we need to convert m2 to cm2 by recalling that

$100cm=1m\phantom{\rule{0ex}{0ex}}{\left(100cm\right)}^{2}={\left(1m\right)}^{2}\phantom{\rule{0ex}{0ex}}10000c{m}^{2}=1{m}^{2}\phantom{\rule{0ex}{0ex}}900{m}^{2}=9000000c{m}^{2}$

Thus,

$\mathrm{Number}\mathrm{of}\mathrm{cement}\mathrm{block}=\frac{9000000c{m}^{2}}{30c{m}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{cement}\mathrm{block}=300000$

Therefore, one would need 300,000 right triangles (5 cm by 12 cm by 13 cm) to cover up a 30 m length square lawn.

## Examples of right triangles problems

A few more problems of right triangles being solved would surely elaborate better.

The figure below comprises two right triangles which are joined together. If the hypotenuse of the bigger right triangle is 15 cm, find the ratio of the area of the bigger to smaller right triangle.

Solution:

Since the length of the hypotenuse of the bigger right triangle is 15 cm, the hypotenuse of the smaller right triangle is

$20cm-15cm=5cm$

We need to find the area of the bigger right triangle, which is Ab, and calculated it as:

$Area=\frac{1}{2}×base×height\phantom{\rule{0ex}{0ex}}{A}_{b}=\frac{1}{2}×9cm×12cm\phantom{\rule{0ex}{0ex}}{A}_{b}=\frac{1}{\overline{)2}}×9cm×{}^{6}\overline{)12}cm\phantom{\rule{0ex}{0ex}}{A}_{b}=9cm×6cm\phantom{\rule{0ex}{0ex}}{A}_{b}=54c{m}^{2}$

Similarly, we need to find the area of the smaller right triangle, which is As, and calculated as

$Area=\frac{1}{2}×base×height\phantom{\rule{0ex}{0ex}}{A}_{s}=\frac{1}{2}×3cm×4cm\phantom{\rule{0ex}{0ex}}{A}_{s}=\frac{1}{\overline{)2}}×3cm×{}^{2}\overline{)4}cm\phantom{\rule{0ex}{0ex}}{A}_{s}=3cm×2cm\phantom{\rule{0ex}{0ex}}{A}_{s}=6c{m}^{2}\phantom{\rule{0ex}{0ex}}$

The ratio of the area of the bigger right triangle Ab to that of the smaller right triangle As is

${A}_{b}:{A}_{s}=54c{m}^{2}:6c{m}^{2}\phantom{\rule{0ex}{0ex}}{A}_{b}:{A}_{s}=\frac{54c{m}^{2}}{6c{m}^{2}}\phantom{\rule{0ex}{0ex}}{A}_{b}:{A}_{s}=\frac{{}^{9}\overline{)54}\overline{)c{m}^{2}}}{{}^{1}\overline{)6}\overline{)c{m}^{2}}}\phantom{\rule{0ex}{0ex}}{A}_{b}:{A}_{s}=\frac{9}{1}\phantom{\rule{0ex}{0ex}}{A}_{b}:{A}_{s}=\mathbf{9}\mathbf{:}\mathbf{1}\phantom{\rule{0ex}{0ex}}$

A right triangle has dimensions 11 cm by 15.6 cm by 11 cm. What type of right triangle is this? Find the perimeter of the right triangle.

Solution:

From the question, since two sides of the right triangle are equal, that means it is an isosceles right triangle.

The perimeter of the right triangle is

$Perimeter=a+b+c\phantom{\rule{0ex}{0ex}}Perimeter=11cm+11cm+15.6cm\phantom{\rule{0ex}{0ex}}Perimeter=\mathbf{37}\mathbf{.}\mathbf{6}\mathbf{}\mathbit{c}\mathbit{m}$

## Right Triangles - Key takeaways

• A right triangle is a triangle in which one angle is a right angle, that is a 90-degree angle.
• The scalene and isosceles right triangles are the two types of right triangles.
• The right triangle consists of three sides, a complementary pair of angles, and a right angle.
• The perimeter of a right triangle of the sum of all the sides.
• The area of the right triangle is the product of half of its base and its height.

#### Flashcards in Right Triangles 19

###### Learn with 19 Right Triangles flashcards in the free StudySmarter app

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What is a right triangle?

A right triangle is a triangle in which one angle is a right angle, that is a 90-degree angle.

What is the formula for the perimeter of a right angle?

The perimeter of a right triangle is the sum of all three sides.

How do you find the area of a right triangle?

The area of the right triangle is the product of half of its base and its height.

How do you find the angles of a right triangle?

The angles of a right triangle are found using SOHCAHTOA when at least one of the side lengths is given.

How do you find the hypotenuse of a right triangle?

In order to find the hypotenuse of a right triangle, you use the Pythagorean theorem, that is you add the squares of each of the base and height, then you take the positive square root of the answer.

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