Question Video: The Right-Angled Triangle Altitude Theorem Mathematics

What does (𝐴𝐢)² equal to?

03:18

Video Transcript

What does 𝐴𝐢 squared equal to?

We’re asked what the square of the length of side 𝐴𝐢 in the triangle shown is equal to. And to work this out, we’re going to use areas. In fact, we’ll see that this is actually one-half of the Euclidean theorem for right triangles.

We begin by redrawing triangle 𝐴𝐡𝐢, adding a square to each of its sides, and labeling the vertices of these squares as shown. Our next step is to project 𝐴 onto side 𝐺𝐻, meeting it at point 𝐾, and then adding the lines 𝐡𝐸 and 𝐴𝐺 to the diagram as shown. We want to now show that triangles 𝐡𝐢𝐸 and 𝐺𝐢𝐴 are congruent. And from this, via equating some areas, we’ll find an expression for 𝐴𝐢 squared.

The first thing we can note is that angles 𝐡𝐢𝐸 and 𝐺𝐢𝐴 are congruent. This is because they’re both right angles added to angle 𝐴𝐢𝐡. Now, we also know that side 𝐢𝐡 is equal to side 𝐢𝐺 and that side 𝐴𝐢 is equal to side 𝐢𝐸. So we see that triangles 𝐡𝐢𝐸 and 𝐺𝐢𝐴 are congruent since they have at least two congruent sides with the included angle between those sides also congruent.

Now, we know that the area of any triangle is half the base times the perpendicular height. So, now applying this to triangle 𝐺𝐢𝐴, where we choose 𝐺𝐢 as the base and where 𝐢𝐷 is the perpendicular height, we have that the area of triangle 𝐺𝐢𝐴 is equal to one-half of 𝐺𝐢 times 𝐢𝐷. We know also that 𝐺𝐢 times 𝐢𝐷 is equal to the area of the rectangle 𝐢𝐷𝐾𝐺. So we have the area of triangle 𝐺𝐢𝐴 equals half the area of the rectangle 𝐢𝐷𝐾𝐺.

So, now making some space and making a note of this, we can look similarly at triangle 𝐡𝐢𝐸. This time, we choose our base as side 𝐸𝐢, and the perpendicular height is 𝐢𝐴. Then, the area of triangle 𝐡𝐢𝐸 equals one-half 𝐸𝐢 times 𝐢𝐴. Now we note that 𝐸𝐢 times 𝐢𝐴 is the area of the square 𝐴𝐢𝐸𝐹. And we can observe also that since triangles 𝐺𝐢𝐴 and 𝐡𝐢𝐸 are congruent, they must have the same area. This, in turn, means that the rectangle 𝐢𝐷𝐾𝐺 must have the same area as the square 𝐴𝐢𝐸𝐹.

Now, the area of rectangle 𝐢𝐷𝐾𝐺 equals 𝐢𝐷 times 𝐷𝐾. And 𝐷𝐾 equals 𝐢𝐺, which is equal to 𝐢𝐡. So the area of rectangle 𝐢𝐷𝐾𝐺 is equal to 𝐢𝐷 times 𝐢𝐡. Similarly, the area of the square 𝐴𝐢𝐸𝐹 equals 𝐴𝐢 times 𝐸𝐹. And since 𝐸𝐹 equals 𝐴𝐢, this square’s area is equal to 𝐴𝐢 squared. So we have the area of the square 𝐴𝐢𝐸𝐹 equal to the area of the rectangle 𝐢𝐷𝐾𝐺. And so, 𝐴𝐢 squared is equal to 𝐢𝐷 multiplied by 𝐢𝐡, which is actually one part of the Euclidean theorem for right triangles.

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