Lesson Video: Right Triangle Altitude Theorem | Nagwa Lesson Video: Right Triangle Altitude Theorem | Nagwa

# Lesson Video: Right Triangle Altitude Theorem Mathematics

In this video, we will learn how to use the right triangle altitude theorem, also known as the Euclidean theorem, to find a missing length.

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### Video Transcript

In this video, weβll 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 allows us to determine the lengths of sides in a right triangle given two of the lengths. Our plan of action in this video is we begin with a right triangle, adding squares to each side of the triangle, and use projections and the properties of congruent triangles to derive the Euclidean theorem. Weβll then use this theorem, together with the Pythagorean theorem, to derive a corollary. Weβll then see how the theorem and the corollary can be applied in some examples.

To derive the Euclidean theorem, we begin with a right triangle π΄π΅πΆ with right angle at π΄. We then project π΄ onto the side πΆπ΅ and call this point π· as shown. The line segment π΄π· is perpendicular to side πΆπ΅. We now add to the diagram three squares given by each side of triangle π΄π΅πΆ and label the vertices of these squares as shown. We next want to continue our projection from π΄ to the side πΆπ΅ down to the side πΊπ». And we then add the line πΆπΉ and π΄πΊ to our diagram as shown. And we want to show now that triangle πΆπ΅πΉ and triangle πΊπ΅π΄ are congruent. We can do this by first noting that the measure of angle πΆπ΅πΉ is equal to the measure of angle π΄π΅πΊ. And this is true because these are both right angles added to angle π΄π΅πΆ.

Now we know that π΅πΆ is equal to π΅πΊ since theyβre the sides of a square and similarly that π΄π΅ is equal to π΅πΉ. And since our triangles have two sides of the same length, including the same angle measure, this is enough to show that the triangles are congruent.

Next, we know that the area of a triangle is half the length of the triangleβs base multiplied by the triangleβs perpendicular height. Applying this to our triangle πΊπ΅π΄, which is an obtuse triangle, and choosing π΅πΊ as the base, we note that π·π΅ is actually our perpendicular height. So the area of triangle π΄π΅πΊ is one over two multiplied by π΅πΊ times π΅π·. Thatβs one over two times the base times the height. Now we know that π΅πΊ times π΅π· is the area of the rectangle π΅π·πΎπΊ. Hence, the area of our triangle π΄π΅πΊ is one over two times the area of the rectangle π΅π·πΎπΊ.

Similarly, we can see choosing the base of triangle π΅πΆπΉ to be π΅πΉ that the area of our triangle π΅πΆπΉ is one over two multiplied by π΅πΉ times π΄π΅. Thatβs where π΄π΅ is the perpendicular height of the triangle. And we know that in this case π΅πΉ multiplied by π΄π΅ is the area of the square π΄πΈπΉπ΅ so that the area of our triangle π΅πΆπΉ is one over two times the area of the square π΄πΈπΉπ΅. Now, recalling that our two triangles π΄π΅πΊ and π΅πΆπΉ are congruent, they must have the same area. And this means that the square π΄πΈπΉπ΅ must have the same area as the rectangle π΅π·πΎπΊ.

So now, if we equate the expressions for their areas, thatβs π΅π΄ squared is equal to π΅π· multiplied by π·πΎ. π΅π΄ squared is the area of our square, and π΅π· multiplied by π·πΎ is the area of the rectangle, noting further that the side π·πΎ has the same length as the side πΆπ» and that πΆπ» has the same length as πΆπ΅. Since theyβre the sides of a square, we have that π΅π΄ squared is equal to π΅π· multiplied by π΅πΆ. Making a note of this, in the same way, we can show that πΆπ΄ squared is equal to πΆπ· multiplied by πΆπ΅. So now making some space, we can summarize our results in the Euclidean theorem. This says if triangle π΄π΅πΆ is a right triangle at π΄ with a projection to π· as shown, then π΅π΄ squared is equal to π΅π· multiplied by π΅πΆ and πΆπ΄ squared is equal to πΆπ· multiplied by πΆπ΅.

We can also use the Euclidean theorem, together with the Pythagorean theorem, to show another useful result. Now recalling that the Pythagorean theorem tells us that the hypotenuse of a right angle triangle squared is equal to the sum of the squares of the other two sides. For our triangle π΄π΅π·, this gives us π΄π· squared plus π΅π· squared is π΄π΅ squared. And subtracting π΅π· squared from both sides, we have π΄π· squared is π΄π΅ squared minus π΅π· squared. Now, using our Euclidean theorem, we substitute π΄π΅ squared, that is, π΅π΄ squared, which is π΅π· multiplied by π΅πΆ. So we have π΄π· squared is equal to π΅π· multiplied by π΅πΆ minus π΅π· squared. And taking out a common factor of π΅π·, on our right-hand side, we have π΅π· multiplied by π΅πΆ minus π΅π·.

And noting that the side length π΅πΆ is the same as π΅π· plus πΆπ·, subtracting π΅π· from both sides gives us πΆπ· is equal to π΅πΆ minus π΅π·. And since π΅πΆ minus π΅π· is the second term on our right-hand side, we have π΄π· squared is π΅π· multiplied by πΆπ·. And so our corollary to the Euclidean theorem is that if π΄π΅πΆ is a right triangle at π΄ with projection to π· as shown, then π΄π· squared is π΅π· multiplied by πΆπ·.

Itβs worth noting that thereβs an alternative and equivalent way of looking at these two results. This is the Euclidean leg rule, where again we have a right triangle π΄π΅πΆ with right angle at π΄. And this is projected onto the hypotenuse πΆπ΅ to the point π·. And this tells us if the leg and the part are as shown, then the hypotenuse divided by the leg is equal to the leg divided by the part. That is, π΅πΆ over π΅π΄ is equal to π΅π΄ over π΅π·. And if we multiply both sides by π΅π΄ and π΅π·, this gives us the first part of our Euclidean theorem: π΅πΆ multiplied by π΅π· is π΅π΄ squared.

Now, if we change our leg and part to the triangle π΄π·πΆ as opposed to π΄π·π΅, again we have hypotenuse over leg is equal to leg over part. That is, π΅πΆ over πΆπ΄ is equal to πΆπ΄ over πΆπ·. And we have the second part of our Euclidean theorem: π΅πΆ multiplied by πΆπ· is πΆπ΄ squared. And similarly, for our second result, the corollary, also called the altitude rule, if the altitude is the perpendicular height π΄π· and the hypotenuse is split into left and right parts, thatβs πΆπ· and π·π΅, we have left over altitude is equal to altitude over right. And this gives us our corollary πΆπ· multiplied by π΅π· is π΄π· squared. And what this tells us is that the altitude or height of the right triangle squared is the product of the lengths of the segments of the hypotenuse when itβs split by the altitude.

So now letβs look at an example of applying the Euclidean theorem to find a missing side length in a right triangle.

Find the length of the line segment π΄π·.

To find the length π΄π·, we first note that π· is the projection of π΅ onto π΄πΆ and that the triangle π΄π΅πΆ is a right triangle at π΅. And now recalling the Euclidean theorem, this tells us that π΄π΅ squared is π΄π· multiplied by π΄πΆ. That is, the side length π΄π΅ squared is the product of the segment π΄π· with the length of the hypotenuse π΄πΆ. Weβre given that π΄π΅ is 34 centimeters and that the hypotenuse π΄πΆ is 40 centimeters. And substituting these values into the Euclidean theorem, this gives us 34 squared is π΄π· multiplied by 40. Dividing through by 40 and rearranging, this gives us π΄π· is 34 squared over 40, that is, 1156 divided by 40, which is 28.9. The line segment π΄π· is therefore 28.9 centimeters.

Letβs now see another example of applying the Euclidean theorem to determine the length of a side in a right triangle.

Determine the length of line segment π΄π·.

We note that π· is the projection of π΄ onto the line π΅πΆ and that the triangle π΄π΅πΆ is a right triangle at π΄. And we recall that the corollary to the Euclidean theorem, that is, the altitude rule, tells us that π΄π· squared is equal to π΅π· multiplied by πΆπ·. This tells us that the altitude or height of the right triangle squared is the product of the lengths of the segments of the hypotenuse when it is split by the altitude.

Weβre given that the two segments are 2.5 centimeters, thatβs πΆπ·, and 6.4 centimeters, thatβs π΅π·. And substituting these values in, we have π΄π· squared is 6.4 multiplied by 2.5. This evaluates to 16, so π΄π· squared is 16. Now, taking the square root on both sides of the equation, noting that π΄π· is a length and so is nonnegative, we get π΄π· is the square root of 16, which is four. The length of π΄π· is therefore four centimeters.

In our next example, weβll apply both parts of the Euclidean theorem to determine a missing side length in a right triangle.

In the figure shown, if ππΏ is equal to 40 and ππΏ is 30, what is the length of ππ?

We note first that πΏ is the projection of π onto ππ. And the triangle πππ is a right triangle at π. We can then recall that the corollary to the Euclidean theorem tells us that ππΏ squared is πΏπ multiplied by ππΏ. Now, since we know the lengths of ππΏ and ππΏ, that is, 40 and 30 centimeters, respectively, this will allow us to find πΏπ. We can then apply the Euclidean theorem to find ππ. So now, substituting in the given lengths to the corollary of the Euclidean theorem, which is also known as the altitude rule, we have 30 squared is πΏπ multiplied by 40. Dividing through by 40 and rearranging, this gives us πΏπ is equal to 30 squared divided by 40, that is, 900 divided by 40, which is 22.5.

So making a note of this and clearing some space, we can now use the Euclidean theorem to note that ππ squared is equal to πΏπ multiplied by ππ. Remember, πΏπ is the shortest segment of the hypotenuse, which is 22.5 centimeters, and ππ is the length of the hypotenuse of the triangle πππ. We know that the length of the hypotenuse ππ is the sum of the two segments, that is, 40 plus 22.5. And thatβs equal to 62.5. Now, substituting these values into our Euclidean theorem, thatβs ππ squared is equal to πΏπ multiplied by ππ, we have πΏπ is 22.5 multiplied by ππ, which is 62.5. And this evaluates to 1406.25. And now taking the positive square root on both sides, since ππ is a length and so must always be positive, we have that ππ is 37.5 centimeters.

Itβs worth noting that we could also have found this length using the Pythagorean theorem with the right triangle ππΏπ. This gives us ππ squared is equal to πΏπ squared plus ππΏ squared. That is, ππ squared is 22.5 squared plus 30 squared, which is 1406.25 as before. And so again, taking the positive square root, we have ππ is 37.5. And so the length ππ is 37.5 centimeters.

In our final example, weβll apply the Euclidean theorem and the Pythagorean theorem to determine the area of the right triangle from a given diagram.

Calculate the area of triangle π΄π΅π·.

We begin by recalling that the area of a triangle is half the length of the triangleβs base multiplied by the triangleβs perpendicular height. And since π΄π΅π· is a right triangle at π·, we can choose π΅π· as the base and then π΄π· as the perpendicular height. The area of the triangle is then one over two times π΅π· multiplied by π΄π·. We can find the length π΄π· using the Euclidean theorem and then the length π΅π· using the Pythagorean theorem.

If we first note that π· is the projection of π΅ onto the side π΄πΆ and that the triangle π΄π΅πΆ is a right triangle at π΅, then the Euclidean theorem tells us that π΄π΅ squared is equal to π΄πΆ multiplied by π΄π·. Substituting now π΄π΅ is 44 centimeters and π΄πΆ is 55 centimeters, we have 44 squared is 55 multiplied by π΄π·. And dividing through by 55, we have 44 squared over 55 is π΄π·, which evaluates to 35.2 centimeters. And we now have the lengths of two sides of our right triangle π΄π΅π·. And so we can use the Pythagorean theorem to find a third length, that is, the length π΅π·.

Applied to our triangle, the Pythagorean theorem tells us that π΄π΅ squared is π΄π· squared plus π΅π· squared. And substituting π΄π΅ is 44 centimeters and π΄π· is 35.2 centimeters, we have 44 squared is 35.2 squared plus π΅π· squared. And subtracting 35.2 squared from both sides and rearranging, this gives us π΅π· squared is 44 squared minus 35.2 squared. Our right-hand side evaluates to 696.96. And taking the positive square root on both sides, since π΅π· is a length, thatβs the square root of 696.96, which is equal to 26.4. So π΅π· is 26.4 centimeters.

So now, we can substitute π΄π· is 35.2 centimeters and π΅π· is 26.4 centimeters into our formula for the area. This gives us the area of triangle π΄π΅π· is one over two times 26.4 multiplied by 35.2. This evaluates to 464.64, and hence the area of triangle π΄π΅π· is 464.64 square centimeters.

Letβs now complete this video by recapping some of the key points weβve covered. The Euclidean theorem tells us that if π΄π΅πΆ is a right triangle at π΄ with projection to π· on the line πΆπ΅, then π΅π΄ squared is π΅π· multiplied by π΅πΆ and πΆπ΄ squared is πΆπ· multiplied by πΆπ΅. There is a useful corollary to the Euclidean theorem that gives us π΄π· squared is equal to π΅π· multiplied by πΆπ·. And this is also called the altitude rule. Sometimes we need to apply both the Euclidean theorem and its corollary. We may also need to apply these in conjunction with the Pythagorean theorem to solve problems.