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10. Right Triangles & Trigonometry
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Math II

10.01 The Pythagorean theorem and its converse

Lesson
Practice
Lesson

Review the Pythagorean theorem

 

A long time ago, a Greek mathematician named Pythagoras came up with an interesting observation about right triangles: if $c$c represents the length of the hypotenuse, and $a$a and $b$b represent the other two sides (that meet at a right angle), then $a^2+b^2=c^2$a2+b2=c2.

In other words, the square of the hypotenuse ($c$c) of a right triangle is equal to the square of side $a$a plus the square of side $b$b.

The following interactive demonstrates using areas Pythagoras' Theorem.  You can use the slider and move the squares of a and b to cover the area of c squared, thus demonstrating that $a^2+b^2=c^2$a2+b2=c2

(Watch this video to see the interactive in action -  )

Created with Geogebra

 

 
Pythagorean theorem

If $\triangle ABC$△ABC is a right triangle, then

$a^2+b^2=c^2$a2+b2=c2,

 where $c$c is the hypotenuse of the triangle. 

 

Practice questions

Question 1

Calculate the value of $c$c in the triangle below.

A right-angled triangle with a right angle shown at the bottom right corner. The base is labeled as 14 cm, the height on the right side is labeled as 48 cm, and the hypotenuse is labeled with c cm. There is a small square at the right angle indicating the 90-degree angle.

Question 2

Calculate the value of $b$b in the triangle below.

 

A right triangle is depicted with the right angle located at the lower left corner. The vertical leg of the triangle is labeled "$10$10 m" and the hypotenuse is labeled "$26$26 m". The base, which runs horizontally along the bottom of the triangle, is labeled "$b$b m", suggesting a length in meters that is not specified. The lengths of the sides are indicative of a measurement in meters. A small square at the junction of the base and the vertical leg signifies the right angle.

Question 3

$VUTR$VUTR is a rhombus with perimeter $112$112 cm. The length of diagonal $RU$RU is $46$46 cm.

A rhombus with vertices labeled U, V, W, and T is illustrated. Dotted lines are drawn from each vertex to the opposite vertex, forming two intersecting diagonals inside the quadrilateral. The point of intersection of the diagonals is marked with a circle and labeled W. Additionally, a small solid blue square is shown, indicating a right angle at vertex W, suggesting that the diagonals are perpendicular to each other. The sides UV and WT are marked with double tick marks, indicating they are of equal length, as are sides VU and RT.

  1. First find the length of $VR$VR.

  2. Then find the length of $RW$RW.

  3. If the length of $VW$VW is $x$x cm, find $x$x correct to two decimal places.

  4. Hence, what is the length of the other diagonal $VT$VT correct to two decimal places.

 

 

The converse of the Pythagorean theorem

Usually we use the Pythagorean theorem to find the sides lengths of a triangle that we already know is a right triangle. That is, we know that if a given right triangle has shorter side lengths of $a$a and $b$b, along with a hypotenuse of length $c$c, then 

$c^2=a^2+b^2$c2=a2+b2

But, we can also use the converse of the Pythagorean theorem to find out for ourselves whether a particular triangle is a right triangle or not. The converse of the theorem says that if the side lengths of a triangle satisfy the above equation, then the triangle must have a right angle.

 

Converse of the Pythagorean theorem 

If $\triangle ABC$△ABC has a longest side with length $c$c and the other two sides have lengths $a$a and $b$b and

$c^2=a^2+b^2$c2=a2+b2

then the triangle is a right triangle. Otherwise, the triangle is not a right triangle.

 

Obtuse and Acute Triangles

When the converse of the Pythagorean Theorem tells us that a triangle is not a right triangle, that triangle must either have:

  • one obtuse angle (which means it is an obtuse triangle), or 
  • all acute angles (which means it is an acute triangle).

Just like with right and non-right triangles, we want to be able to tell these two types of triangles apart just by looking at the side lengths of a triangle.

 

Investigate: Side lengths and angles

Again we can do this by thinking about the longest side in a triangle. First consider what happens when you change the position of any two sides of a triangle without changing their lengths. What happens to the length of the remaining side? What happens to the angle opposite the remaining side (which is between the two fixed ones)? For example, think about what happens to the triangle formed by the two hands of a clock as the hands change their position.

As the two sides move closer together, the remaining side length will get smaller and so will the size of the angle opposite it. This reflects the fact that an angle of a triangle is related to the size of the side length opposite it.

In particular, the longest side length will always be opposite the largest angle in the triangle. So, if you increase the length of a triangle's longest size, the largest angle will be bigger. Likewise, if the longest side gets shorter then the largest angle will be smaller.

So if the longest side is long enough, then the largest angle will be obtuse (and so will the triangle), If the longest side is short enough, then the largest angle will be acute (and so will the triangle). If the longest side is just the right size (that is, it satisfies the Pythagorean Theorem), then the largest angle with be a right angle and the triangle will be a right triangle.

We can summarize all of these facts, therefore, by comparing the size of the longest side in a triangle in terms of the Pythagorean Theorem.

Pythagorean results

If a triangle has a longest side with length $c$c and the other two sides have lengths $a$a and $b$b, then we have the following results:

  • If $c^2>a^2+b^2$c2>a2+b2 then the triangle is an obtuse triangle.
  • If $c^2=a^2+b^2$c2=a2+b2 then the triangle is a right triangle.
  • If $c^2c2<a2+b2 then the triangle is an acute triangle.

 

Review: Triangle inequality

The advantage of the above results is that you don't actually need to see a triangle to know what it looks like. You can get all of this information just from the numbers that indicate side lengths. However, this can also be a problem, since it might be the case that a certain set of side lengths can't actually form any triangle at all.

To check whether three lengths can form a triangle at all, we first need to check that they satisfy the triangle inequality.

The Triangle Inequality

For any triangle, the sum of any two side lengths must be greater than the remaining side length.

That is, if $a$a, $b$b and $c$c are all side lengths in a triangle, then $a+b>c$a+b>c.

 

Practice questions

QUESTION 4

Use Pythagoras' theorem to determine whether this is a right triangle.

A triangle with labeled side lengths. The triangle has sides with different lengths and with no implicit indication of angle measurements. The sides are measured with lengths of $10$10 units, $18$18 units and $21$21 units.

  1. Let $a$a and $b$b represent the two shorter side lengths. First find the value of $a^2+b^2$a2+b2.

  2. Let $c$c represent the length of the longest side. Find the value of $c^2$c2.

  3. Is the triangle a right triangle?

    Yes

    A

    No

    B

Question 5

Consider a triangle whose shortest sides have lengths $6$6 and $8$8. The longest side of the triangle has a length of $c$c.

  1. What must the value of $c$c be if the triangle has a right angle?

  2. If $c=12$c=12, then which of the following correctly describes the triangle?

    Right angled

    A

    Obtuse

    B

    Acute

    C

  3. If $c=9$c=9, then which of the following correctly describes the triangle?

    Acute

    A

    Obtuse

    B

    Right angled

    C

QUESTION 6

Consider three straight lines with lengths $9$9, $12$12 and $14$14 units.

  1. Is it possible to form a triangle using these lines?

    Yes

    A

    No

    B

  2. Which of the following correctly describes the triangle formed?

    Obtuse triangle

    A

    Acute triangle

    B

    Right Triangle

    C

  3. Which of the following inequalities justifies your answer in part (b)?

    $9^2+12^2>14^2$92+122>142

    A

    $9+12<14$9+12<14

    B

    $9+12>14$9+12>14

    C

    $9^2+12^2<14^2$92+122<142

    D

Outcomes

II.G.SRT.8

Use trigonometric ratios and the pythagorean theorem to solve right triangles in applied problems.

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