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Apply Pythagoras' theorem in two-dimensional contexts

Start a set of 8
Step 3 of 8

Find the triangle before calculating

The heights of two objects are not usually the sides of the required triangle. Use the separation between the endpoints. You already know how to find a side once the right triangle is identified

  1. 1Locate the right angle Find the two perpendicular segments. Vertical supports and level ground make a right angle
  2. 2Find the relevant separation For tops measured from the same level ground, subtract the shorter height from the taller height
  3. 3Choose the side direction Add squared legs for a hypotenuse. For an unknown leg, subtract the known leg's square from the hypotenuse's square \(\displaystyle \begin{aligned}c^2&=a^2+b^2\\a^2&=c^2-b^2\end{aligned}\)
  4. 4Finish in the requested units Take the positive square root. Keep the calculator's full value until the final rounding
Year 9, Chapter 3, 4 difficulty levels, Calculator allowedBuild a worksheet

Model a flat situation

Locate the endpoints and a right angle. Work out any missing separation before applying Pythagoras' theorem

\[c^2=a^2+b^2\]
  1. Use the relevant part A separation can be a difference of heights or a remaining section of a total. Use lengths measured in the same direction from a stated reference
  2. Identify the hypotenuse It is opposite the right angle. A straight distance is not automatically a hypotenuse in every triangle
  3. Find the requested side Add squared legs to find a squared hypotenuse. Subtract the known leg's square from the hypotenuse's square to find a squared leg \(\displaystyle \begin{aligned}c&=\sqrt{a^2+b^2}\\a&=\sqrt{c^2-b^2}\end{aligned}\)
  4. Link triangles carefully Check the right angle in each triangle. Carry an exact squared shared side forward and round only the requested final length

Worked example

Two vertical supports stand on level ground. Their heights are 12 m and 5 m. A taut line between their tops is 25 m long. Find the distance between their bases

\[x>0\]
  1. \(\displaystyle h=12-5\)Subtract heights measured from the same ground level
  2. \(\displaystyle h=7\text{ m}\)This is the vertical separation between the tops
  3. \(\displaystyle x^2+7^2=25^2\)The taut line is the hypotenuse. The horizontal and vertical separations are the legs
  4. \(\displaystyle x^2+49=625\)Evaluate the known squares
  5. \(\displaystyle x^2+49-49=625-49\)Subtract the known leg's square from both sides
  6. \(\displaystyle x^2=576\)This is the square of the required distance
  7. \(\displaystyle x=\sqrt{576}\)Take the positive root because a distance is positive
  8. \(\displaystyle x=24\text{ m}\)The distance between the bases is exact

Common mistakes

Adding heights from the same ground level
✕\(\displaystyle \begin{aligned}h&=15+8\\h&=23\text{ m}\end{aligned}\)✓\(\displaystyle \begin{aligned}h&=15-8\\h&=7\text{ m}\end{aligned}\)
Two vertical supports stand on the same level ground, with heights 15 m and 8 m. Their tops' vertical separation h is the difference of those heights
Adding squares for an unknown leg
✕\(\displaystyle x^2=25^2+7^2\)✓\(\displaystyle x^2=25^2-7^2\)
A right triangle has hypotenuse 25 m, known leg 7 m and unknown leg x. Subtract the known leg's square from the hypotenuse's square to find the other leg's square
Rounding a shared side too soon
✕\(\displaystyle \begin{aligned}AC&\approx8.1\\AC^2&\approx65.61\end{aligned}\)✓\(\displaystyle \begin{aligned}AC^2&=4^2+7^2\\AC^2&=16+49\\AC^2&=65\end{aligned}\)
Triangle ABC has a right angle at B, AB is four metres and BC is seven metres. AC is shared with a second right triangle. Re-squaring the rounded AC changes its squared value; carry the exact squared length into the second triangle

Difficulty levels

introFind an endpoint separation, then an exact whole-number distance
standardInterpret a flat diagram and find either a hypotenuse or a shorter side. Round only when requested
challengeUse two linked right triangles in one plane and keep the shared squared side exact
examSelect a final length for the same linked-triangle structures, with the stated precision

Questions about this topic

When should I subtract two stated lengths?+

When they share a direction and reference and the question asks for the gap between them. Check the endpoints before subtracting

How do I start when there are two triangles?+

Start with the triangle that has two known sides. Its result supplies the shared side for the other triangle

Should I round the shared side?+

No. Keep its exact square or the full calculator value until you have the final requested length

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 3: Pythagoras and trigonometry

In this chapter

3A Finding the hypotenuse: Find a side of a right-angled triangle3B Finding a shorter side: Find a side of a right-angled triangleApply Pythagoras' theorem in two-dimensional contextsPythagoras in three dimensions (optional extension)Name sides and state trigonometric ratiosChoose a trigonometric ratio3F Unknown side in the numerator: Evaluate a trigonometric ratio3F Unknown side in the numerator: Find an unknown side of a right-angled triangle3G Unknown side in the denominator: Find a side in the denominator3H Finding angles with trigonometry: Find an angle from a trigonometric ratioFind an angle of a right-angled triangle from two sidesAngles of elevation and depressionSolve a practical right-angled-triangle problemRead and state three-figure bearingsSolve a navigation problem with bearings