Finding a missing side and angle in a right triangle

Key takeaway: Use trig ratios and inverse trig to connect the given sides and angle to the unknown values.

Identify the known information

This problem asks for two different unknowns: a side length $x$ and an angle $y$. Because the figure is a right triangle, the key is to use the given measurements to choose a trig ratio that fits each unknown.

A right triangle problem usually becomes easier when you decide whether the known sides are opposite, adjacent, or the hypotenuse relative to the marked angle. That choice tells you whether to use sine, cosine, or tangent.

Find the missing side

If the given sides are $8\text{ cm}$ and $9\text{ cm}$ and they form the two legs of the right triangle, then the Pythagorean theorem gives the third side:

$x^2 = 8^2 + 9^2$ $x^2 = 64 + 81 = 145$ $x = \sqrt{145} \approx 12.0\text{ cm}$

So the missing side is

$\boxed{x\approx 12.0\text{ cm}}$

Find the angle

Now use a trig ratio to find $y$. If $y$ is the angle opposite the 8 cm side and adjacent to the 9 cm side, then

$\tan y = \frac{8}{9}$

Taking inverse tangent:

$y = \tan^{-1}\left(\frac{8}{9}\right) \approx 41.6^\circ$

Rounded to the nearest degree:

$\boxed{y\approx 42^\circ}$

Why this works

The Pythagorean theorem is the best tool for a missing side when two legs are known. Once the side length is found, the angle can be found from a trig ratio. In many triangle questions, one part uses geometry and the other uses trigonometry, so the order matters.

Final check

A triangle with legs 8 and 9 should have a hypotenuse a little bigger than 12, which matches the result. Also, the angle opposite the shorter leg should be smaller than the angle opposite the longer leg, so an answer near $42^\circ$ is reasonable.

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Pitfalls the pros know 👇 A common mistake is to assume the given 8 cm and 9 cm are always the hypotenuse and one leg. Before using the Pythagorean theorem, check whether the diagram shows a right angle between those two sides. Another error is using the wrong side ratio for the angle: if 8 cm and 9 cm are the legs, then tangent is the natural choice for the marked acute angle. Also, do not round $x$ before using it to find $y$ if your teacher expects exact or more accurate intermediate work.

What if the problem changes? If the triangle were changed so that 8 cm was the hypotenuse and 9 cm was one leg, then the Pythagorean theorem would not work in the same way because the numbers would not describe a valid right triangle. If instead the known sides were 8 cm and 15 cm, you would get $x=17$ using $8^2+15^2=17^2$, and then an angle could be found with $\tan^{-1}(8/15)$. The strategy is always the same: identify the triangle structure first, then choose the theorem or ratio that matches it.

Tags: Pythagorean theorem, inverse tangent, hypotenuse