MYP Standard Mathematics Topic Right Triangle…

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MYP Standard Mathematics Topic Right Triangle… | RevisionDojo

Right Triangles Let You Measure The "Unmeasurable"

Right angles appear everywhere: walls meeting floors, books resting on tables, streets meeting at corners.
Because right triangles are so common, a large part of geometry and trigonometry is built around them.
The key idea is that if you know enough information about a right triangle, you can calculate the parts you cannot measure directly (for example, the height of a building or the width of a river).

Definition

Right triangle

A triangle with one angle equal to $90^\circ$.

In right triangle geometry, there are two main tools:

the Pythagorean Theorem, which relates the three side lengths
the trigonometric ratios (sine, cosine, tangent), which connect angles to side ratios

The Pythagorean Theorem Relates The Three Sides

Consider a right triangle with legs (the two shorter sides) of lengths $a$ and $b$, and hypotenuse (the longest side, opposite the right angle) of length $c$.

Definition

Hypotenuse

The side opposite the right angle in a right triangle, always the longest side.

Definition

Pythagorean theorem

In any right triangle, the squares of the two shorter sides add to the square of the hypotenuse: $a^2+b^2=c^2$.

Note

This theorem is powerful because it lets you find a missing side length if you know the other two.

Finding A Missing Side Using Pythagoras

If you know $a$ and $b$, then $$c=\sqrt{a^2+b^2}$$
If you know $c$ and one leg (say $a$), then $$b=\sqrt{c^2-a^2}$$

Example question

A right triangle has legs $6\text{ cm}$ and $8\text{ cm}$. Find the hypotenuse.

Solution

$$\;c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10\text{ cm}$$

Common Mistake

Always check that you are squaring the two legs and that $c$ is the hypotenuse.
A common mistake is to treat a shorter side as $c$, which makes $c^2-a^2$ negative.

The Converse Helps You Test If A Triangle Is Right-Angled

Sometimes you are given three side lengths and need to decide whether the triangle is right-angled.

Definition

Converse

A statement formed by switching the “if” part and the “then” part of a conditional statement.

The converse of the Pythagorean Theorem is true:

If the side lengths of a triangle satisfy $a^2+b^2=c^2$ (where $c$ is the longest side), then the triangle is a right triangle.

This gives a practical test:

identify the longest side and call it $c$
compute $a^2+b^2$ and compare with $c^2$

Acute Or Obtuse? Comparing $a^2+b^2$ With $c^2$

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Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

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MYP Standard Mathematics Right Triangle Geometry Notes

Right Triangles Let You Measure The "Unmeasurable"

The Pythagorean Theorem Relates The Three Sides

Finding A Missing Side Using Pythagoras

The Converse Helps You Test If A Triangle Is Right-Angled

Acute Or Obtuse? Comparing $a^2+b^2$ With $c^2$

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Right Triangles Let You Measure The "Unmeasurable"

The Pythagorean Theorem Relates The Three Sides

Finding A Missing Side Using Pythagoras

The Converse Helps You Test If A Triangle Is Right-Angled

Acute Or Obtuse? Comparing $a^2+b^2$ With $c^2$

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

MYP Standard Mathematics Right Triangle Geometry Notes

Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

Right Triangles Let You Measure The "Unmeasurable"

The Pythagorean Theorem Relates The Three Sides

Finding A Missing Side Using Pythagoras

The Converse Helps You Test If A Triangle Is Right-Angled

Acute Or Obtuse? Comparing $a^2+b^2$ With $c^2$

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Right Triangles Let You Measure The "Unmeasurable"

The Pythagorean Theorem Relates The Three Sides

Finding A Missing Side Using Pythagoras

The Converse Helps You Test If A Triangle Is Right-Angled

Acute Or Obtuse? Comparing $a^2+b^2$ With $c^2$

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account