In this explainer, we will learn how to use the right triangle altitude theorem, also known as the Euclidean theorem, to find a missing length.
This theorem is a useful tool to rewrite expressions involving the lengths of sides in a right triangle with a projection from the right angle onto the hypotenuse. In particular, it will allow us to determine the lengths of sides in a right triangle given two of the lengths.
Letβs start by deriving the Euclidean theorem. To do this, we first construct a right triangle with the right angle at . We then project onto and call this point as shown.
We now add to the diagram three squares given by each side of and label the vertices of these squares as shown.
We now want to project onto and add the lines and to the diagram as shown.
We now want to show that and are congruent. We do this by first noting that since these are both right angles added to . We know that and that . This gives us the following.
This is enough to show the triangles are congruent.
Next, we know that the area of a triangle is half the length of the triangleβs base multiplied by the triangleβs perpendicular height. Applying this to , we choose as the base and we note that is the perpendicular height as shown.
Thus,
We know that is the area of . So,
Similarly, we can see that
We also note that
So,
Finally, since these triangles are congruent, they have the same area, which means that square has the same area as rectangle . We can then equate the expressions for their areas:
We note that and that , so we can rewrite this as
We can show a similar result for
We can summarize the result we have shown as follows.
Theorem: Euclidean Theorem
In any right triangle, the area of the square on a side adjacent to the right angle is equal to the area of the rectangle whose dimensions are the length of the projection of this side on the hypotenuse and the length of the hypotenuse.
In general, if is a right triangle at with a projection to as shown, then
We can use the Euclidean theorem together with the Pythagorean theorem to show another useful result. First, the Pythagorean theorem gives us
Next, we can use the Euclidean theorem to substitute into this equation. We get
We can then take out the shared factor of on the right-hand side of the equation to get
Finally, we know that , so . Hence,
We can write this result formally as follows.
Theorem: Corollary to the Euclidean Theorem
If is a right triangle at with projection to as shown, then
Letβs now see some examples of applying the Euclidean theorem to find missing side lengths in right triangles.
Example 1: Finding a Side Length in a Right Triangle Using the Euclidean Theorem
Find the length of .
Answer
We note that is the projection of onto and that is a right triangle at . We can then recall that the Euclidean theorem tells us that
Substituting in the given lengths, we have
We can then divide the equation through by 40 to get
Letβs now see another example of applying the Euclidean theorem to determine the length of a side in a right triangle.
Example 2: Finding the Altitude Using the Hypotenuse Divided into Two Segments
Determine the length of .
Answer
We note that is the projection of onto and that is a right triangle at . We can then recall that the corollary of the Euclidean theorem tells us that
Substituting in the given lengths gives
Taking the square root of both sides of the equation, noting that is a length and so is nonnegative, we get
In our next example, we will apply both parts of the Euclidean theorem to determine a missing side length in a right triangle.
Example 3: Finding an Unknown Length Using Both Parts of the Euclidean Theorem
In the figure shown, if and , what is the length of ?
Answer
We note that is the projection of onto and that is a right triangle at . We can then recall that the corollary to the Euclidean theorem tells us that
Since we know the lengths of and , this will allow us to find . Then, we can apply the Euclidean theorem to find .
Substituting in the given lengths into the corollary of the Euclidean theorem gives us
We divide the equation through by 40 to see that
We can now use the Euclidean theorem to note that
We know that
Substituting and gives
Taking the square root of both sides of the equation, noting that is a length and so is nonnegative, we get
It is worth noting that we could also have found this length using the Pythagorean theorem. Applying the Pythagorean theorem to right triangle , we get
Substituting and gives
Hence,
In our next example, we will apply the Euclidean theorem and Pythagorean theorem to determine a missing length in a right triangle.
Example 4: Finding an Unknown Length Using the Euclidean and Pythagorean Theorems
From the figure, determine the length of . If necessary, round your answer to the nearest hundredth.
Answer
We can start by applying the Pythagorean theorem to right triangle . This gives
Since we know the lengths of and , this will allow us to find . We can then apply the Euclidean theorem to find . Finally, we can then apply the Pythagorean theorem to to find .
Substituting and gives us
Taking the square root of both sides of the equation gives us
We note that is the projection of onto and that is a right triangle at . We can then recall that the Euclidean theorem tells us that
Substituting and gives
Dividing the equation through by 17 then yields
Finally, we apply the Pythagorean theorem to triangle . This gives
Substituting and gives us
We then take the square root of both sides of the equation to get
Rounding to the nearest hundredth gives us 7.06 cm.
In our next example, we will apply the Euclidean theorem and Pythagorean theorem to determine the area of a right triangle from a given diagram.
Example 5: Finding the Area of a Triangle Using the Right Triangle Theorems
Calculate the area of .
Answer
We start by recalling that the area of a triangle is half the length of the triangleβs base multiplied by the triangleβs perpendicular height. Since is a right triangle at , we can choose as the base and then is the perpendicular height, giving us
We can find the length of using the Euclidean theorem and then the length of using the Pythagorean theorem. First, we note that is the projection of onto and that is a right triangle at . The Euclidean theorem then tells us that
We substitute and to get
We then divide the equation through by 55, giving us
We now have the lengths of two sides of right triangle . We can find the third length by applying the Pythagorean theorem, which gives
Substituting and gives
We can then rearrange and simplify the equation:
We then take the square root of both sides of the equation, noting that is a length and so is nonnegative, which gives
We now substitute and into the formula for the area of to get
In our final example, we will apply both the Euclidean and Pythagorean theorems to determine the length of a missing side in a given diagram.
Example 6: Finding an Unknown Length in a Square Divided into Triangles Using Right Triangle Theorems
Find the length of , approximating the result to the nearest hundredth.
Answer
We start by applying the Pythagorean theorem to right triangle . This gives
Since we know the values of and , we can use this to find . We can then use the Euclidean theorem and the values of and to find . Finally, we can apply the Pythagorean theorem to to find .
Substituting and into the above equation gives us
Taking the square root of both sides of the equation gives us
We note that is the projection of onto and that is a right triangle at . We can then recall that the Euclidean theorem tells us that
Substituting and gives
Dividing the equation through by 65 then yields
Finally, we apply the Pythagorean theorem to triangle . This gives
Substituting and gives us
We then take the square root of both sides of the equation to get
Rounding to the nearest hundredth gives us 28.43 cm.
Letβs finish by recapping some of the important points from this explainer.
Key Points
- The Euclidean theorem tells us that if is a right triangle at with projection
to as shown, then
- There is a useful corollary to the Euclidean theorem that we find by applying the Pythagorean; we can also show that
- Sometimes we will need to apply both the Euclidean theorem and its corollary. We may also need to apply these results in conjunction with the Pythagorean theorem to solve problems.
