C6.1; E6.1 - Pythagoras and missing lengths

C6.1; E6.1 - Pythagoras and missing lengths

A diagonal can be shorter than a journey round two sides, but its length is not a guess. You will use the right-angle relationship between three sides, find either missing side, and check whether lengths can belong to a right-angled triangle.

The right-angle relationship

A rectangle is 8 cm8\,\mathrm{cm} wide and 15 cm15\,\mathrm{cm} long. Is its diagonal 23 cm23\,\mathrm{cm}, or something shorter? The diagonal cuts across instead of following both edges. To find its length, compare squares built on the sides of the right-angled triangle.

Hypotenuse

The hypotenuse is the side opposite the 90∘90^\circ angle in a right-angled triangle. It is the longest side. The two sides meeting at the right angle are the shorter sides, however the triangle is turned.

For a right triangle with shorter sides 33 and 44, the hypotenuse has length 55. The squares on those sides have areas 99, 1616 and 2525: the first two areas add to the third. This relationship works for every right triangle, not just whole-number lengths.

[DIAGRAM: asset_name: An area argument for Pythagoras; asset_slug: pythagoras_area_relationship; caption: Subtract four equal triangles from the large square.; description: A square of side a plus b contains four congruent right triangles with perpendicular sides a and b. Their hypotenuses form an inner square of side c. The inner square area is the outer area minus four triangle areas.]
Subtract four equal triangles from the large square.

Here each small triangle has shorter sides aa and bb, and hypotenuse cc. Its two acute angles add to 90∘90^\circ, so the angles of the middle shape are right angles. Its four equal sides have length cc, making it a square. The outer square can also be split into squares of areas a2a^2 and b2b^2, plus two rectangles each of area abab. Subtract the four triangle areas from that total:

c2=(a+b)2−4(12ab)outer area minus triangles=a2+2ab+b2−2absplit the outer area=a2+b2the shared terms cancel\begin{aligned} c^2&=(a+b)^2-4\left(\frac12ab\right) && \text{outer area minus triangles}\\ &=a^2+2ab+b^2-2ab && \text{split the outer area}\\ &=a^2+b^2 && \text{the shared terms cancel} \end{aligned}

This is a relationship between squares of lengths, not a rule for adding the lengths themselves.

Pythagoras' theorem

In a right-angled triangle, with positive shorter side lengths a,ba,b and hypotenuse cc,

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

Not in the List of formulas: learn it.

The reverse test also works: if the squares of the two shorter sides add to the square of the longest side, the triangle is right-angled. Identify the longest side first. If the equality fails, those three lengths do not form a right triangle.

Worked example 1: checking the right-angle relationship

Question. A triangle has side lengths 9 cm9\,\mathrm{cm}, 12 cm12\,\mathrm{cm} and 15 cm15\,\mathrm{cm}. Determine whether it is right-angled.

Plan. Compare the sum of the two smaller squares with the largest square.

92+122=81+144=225square the shorter lengths152=225square the longest length92+122=152the two values agree\begin{aligned} 9^2+12^2&=81+144=225 && \text{square the shorter lengths}\\ 15^2&=225 && \text{square the longest length}\\ 9^2+12^2&=15^2 && \text{the two values agree} \end{aligned}

Answer. The triangle is right-angled, with hypotenuse 15 cm15\,\mathrm{cm}.

Check. Dividing all three lengths by 33 gives 3,4,53,4,5, the right triangle already considered. Scaling lengths preserves angles.

Change only the longest side from 1515 to 1616. Having two familiar lengths does not guarantee a right angle.

Worked example 2: a failed test

Question. A triangle has side lengths 9 cm9\,\mathrm{cm}, 12 cm12\,\mathrm{cm} and 16 cm16\,\mathrm{cm}. Determine whether it is right-angled.

Plan. Make the same comparison, using 1616 as the longest side.

92+122=225the shorter sides are unchanged162=256square the new longest side225≠256the required equality fails\begin{aligned} 9^2+12^2&=225 && \text{the shorter sides are unchanged}\\ 16^2&=256 && \text{square the new longest side}\\ 225&\ne256 && \text{the required equality fails} \end{aligned}

Answer. The triangle is not right-angled.

Check. A right triangle with shorter sides 99 and 1212 has longest side 225=15\sqrt{225}=15, not 1616.

Watch out

a+b=ca+b=c is not Pythagoras. A genuine triangle's two shorter sides together are longer than its longest side. It is their squares that add in the right-angle relationship.

Apply the square test to a new triangle.

Finding a hypotenuse

When the missing side is opposite the right angle, its square is the sum of the other two squares. Find that sum, then take the positive square root. A length cannot be negative.

[DIAGRAM: asset_name: Choosing the missing side; asset_slug: pythagoras_missing_lengths; caption: The side opposite the square corner is the hypotenuse.; description: Left: a right triangle has perpendicular sides fifteen and eight centimetres and hypotenuse c. Right: a right triangle has hypotenuse ten centimetres, one shorter side six centimetres and the other shorter side x. Right angles are marked.]
The side opposite the square corner is the hypotenuse.

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