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PYTHAG - Applications using Pythagoras


Remember Pythagoras' Theorem?

Pythagoras' Theorem

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

  • $c$c is the length of the hypotenuse, and
  • $a$a and $b$b are the lengths of the other two sides

We can rearrange this equation to find formulas for each side length.

Rearranging Pythagoras' Theorem

To find the hypotenuse:        $c=\sqrt{a^2+b^2}$c=a2+b2

 To find a shorter side:     $a=\sqrt{c^2-b^2}$a=c2b2

To apply Pythagoras' Theorem to real life situations,

  1. Look for right-angled triangles
  2. Choose which side, hypotenuse or a shorter side, you are trying to find
  3. Find the lengths of the other two sides
  4. Apply the relevant formula and substitute the lengths of the other two sides

Let's look at some examples so we can see this in action.


Worked Examples

Question 1

Consider a cone with slant height $13$13m and perpendicular height $12$12m.

A cone with a circular base. The cone altitude, illustrated by a vertical dashed line, measures 12 meters, highlighted by a scale line on the left. The radius of the base circle is represented by a horizontal dashed line and is labeled r. These two lines are perpendicular, forming a right angle, which is denoted by a small square symbol. The slant height of the cone, which stretches from the apex to a point on the circumference of the base and opposite to the right angle, measures 13 meters, as indicated by the slanted scale line placed on the right. Together, the radius of the base (base), the altitude of the cone (height), and the slant height (hypotenuse) compose a right-angled triangle.

  1. Find the length of the radius, $r$r, of the base of this cone.

  2. Hence, find the length of the diameter of the cone's base.

Question 2


Find the length of the unknown side, $x$x, in the given trapezium.

Give your answer correct to two decimal places.


A right trapezoid $ABDC$ABDC is depicted as suggested by the two adjacent right angles $\angle BAC$BAC or $\angle CAB$CAB on vertex $9$9 and $\angle DCA$DCA or $\angle ACD$ACD on vertex $7$7. Side $AB$AB or $BA$BA and Side $DC$DC or $CD$CD are the parallel sides of the trapezoid and Side $AB$AB or $BA$BA is longer than side $DC$DC or $CD$CD. Side $CA$CA or $AC$AC measures 9 units and is perpendicular to the two parallel sides. Side $CA$CA or $AC$AC is the base of the figure. Side $AB$AB or $BA$BA is measured as 13 units. Side $DC$DC or $CD$CD is measured as 7 units. Side $BD$BD or $DB$DB is labeled as x units.



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