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mathematics

Right triangle

Right triangle is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Right triangle rather than just read about it. In short: A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1⁄4 turn or 90 degrees). The side opposite to the right angle is called the hypotenuse (side c {\displaystyle c} in the figure).

Right triangle — main illustration
Right triangle — illustration

Key takeaways

  • Right triangle belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Right triangle to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Right triangle from memory before moving on to harder problems.

Reference excerpt

A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1⁄4 turn or 90 degrees). The side opposite to the right angle is called the hypotenuse (side c {\displaystyle c} in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side a {\displaystyle a} may be identified as the side adjacent to angle B {\displaystyle B} and opposite (or opposed to) angle A , {\displaystyle A,} while side b {\displaystyle b} is the side adjacent to angle A {\displaystyle A} and opposite angle B . {\displaystyle B.}

Every right triangle is half of a rectangle which has been divided along its diagonal. When the rectangle is a square, its right-triangular half is isosceles, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is scalene. Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at the apex and the hypotenuse as the base; conversely, the circumcircle of any right triangle has the hypotenuse as its diameter. This is Thales' theorem. The legs and hypotenuse of a right triangle satisfy the Pythagorean theorem: the sum of the areas of the squares on two legs is the area of the square on the hypotenuse, a 2 + b 2 = c 2 . {\displaystyle a^{2}+b^{2}=c^{2}.} If the lengths of all three sides of a right triangle are integers, the triangle is called a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. The relations between the sides and angles of a right triangle provide one way of defining and understanding trigonometry, the study of the metrical relationships between lengths and angles.

Principal properties

Sides

The three sides of a right triangle are related by the Pythagorean theorem, which in modern algebraic notation can be written

a 2 + b 2 = c 2 , {\displaystyle a^{2}+b^{2}=c^{2},}

where c {\displaystyle c} is the length of the hypotenuse (side opposite the right angle), and a {\displaystyle a} and b {\displaystyle b} are the lengths of the legs (remaining two sides). This theorem was proven in antiquity, and is proposition I.47 in Euclid's Elements: "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Three integers ⁠ a {\displaystyle a} ⁠, ⁠ b {\displaystyle b} ⁠, and ⁠ c {\displaystyle c} ⁠ satisfying this equation are called a Pythagorean triple. In practical applications such as construction and surveying, this relationship is frequently applied using the 3-4-5 rule to ensure that an angle is exactly 90 degrees.

Area As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is the height, so the area of a right triangle is one half the product of the two legs. As a formula the area T {\displaystyle T} is

T = 1 2 a b {\displaystyle T={\tfrac {1}{2}}ab}

where a {\displaystyle a} and b {\displaystyle b} are the legs of the triangle. If the incircle is tangent to the hypotenuse A B {\displaystyle AB} at point P , {\displaystyle P,} then letting the semi-perimeter be s = 1 2 ( a + b + c ) , {\displaystyle s={\tfrac {1}{2}}(a+b+c),} we have | P A | = s − a {\displaystyle |PA|=s-a} and | P B | = s − b , {\displaystyle |PB|=s-b,} and the area is given by

T = | P A | ⋅ | P B | = ( s − a ) ( s − b ) . {\displaystyle T=|PA|\cdot |PB|=(s-a)(s-b).}

This formula only applies to right triangles.

Altitudes

If an altitude is drawn from the vertex, with the right angle to the hypotenuse, then the triangle is divided into two smaller triangles; these are both similar to the original, and therefore similar to each other. From this:

The altitude to the hypotenuse is the geometric mean (mean proportional) of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. In equations,

… excerpt ends here. Continue reading the full article.

Illustrations

Right triangle: A right triangle △ABC with its right angle at C, hypotenuse c, and legs a and b,
A right triangle △ABC with its right angle at C, hypotenuse c, and legs a and b,
Right triangle: The Bride's Chair from the proof of the Pythagorean theorem, in the colored version used by Byrne's 1847 edition. The proof shows that the black and yellow areas are equal, as are the red and blue areas.
The Bride's Chair from the proof of the Pythagorean theorem, in the colored version used by Byrne's 1847 edition. The proof shows that the black and yellow areas are equal, as are the red and blue areas.
Right triangle: Altitude f of a right triangle
Altitude f of a right triangle
Right triangle: The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides (p + q, r, s ), (r, p, h ) and (s, h, q ), ( p + q ) 2 = r 2 + s 2 p 2 + 2 p q + q 2 = p 2 + h 2 ⏞ + h 2 + q 2 ⏞ 2 p q = 2 h 2 ∴ h = p q {\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}}
The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides (p + q, r, s ), (r, p, h ) and (s, h, q ), ( p + q ) 2 = r 2 + s 2 p 2 + 2 p q + q 2 = p 2 + h 2 ⏞ + h 2 + q 2 ⏞ 2 p q = 2 h 2 ∴ h = p q {\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}}
Right triangle: Median of a right angle of a triangle
Median of a right angle of a triangle

Worked examples

Example 1 — a first encounter with Right triangle

Start with the simplest possible case. Write down what Right triangle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Right triangle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Right triangle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Right triangle

In research
Right triangle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Right triangle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Right triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Orthogonality, Trigonometry, Types of triangles, so understanding it makes those chapters shorter.
In everyday life
Look for Right triangle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Right triangle in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Right triangle means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Right triangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Right triangle in simple terms?

A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1⁄4 turn or 90 degrees). The side opposite to the right angle is called the hypotenuse (side c {\displaystyle c} in…

Why does Right triangle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Right triangle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Right triangle.

Tags

  • Orthogonality
  • Trigonometry
  • Types of triangles

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