Video Transcript
𝐴𝐵𝐶𝐷 is a square, 𝐴𝐸 equals 6.4, and 𝐸𝐷 equals 6.1. What is the area of the shaded region, to the nearest hundredth?
Let’s look at the given figure more closely. We’re told that 𝐴𝐵𝐶𝐷 is a square. There is then a right triangle, triangle 𝐴𝐷𝐸, within this square, and we’re asked
to find the shaded area, which is all of the area inside the square that is not
taken up by the triangle.
We can therefore find the shaded area by subtracting the area of triangle 𝐴𝐷𝐸 from
the area of the square. We’re given some other lengths in the question. 𝐴𝐸 is 6.4 and 𝐸𝐷 is 6.1. As these are perpendicular sides of the right triangle, this allows us to calculate
its area. The area of a triangle is equal to its base multiplied by its perpendicular height
over two. So the area of triangle 𝐴𝐷𝐸 is 6.1 multiplied by 6.4 over two.
In order to find the area of the square 𝐴𝐵𝐶𝐷, we need to know its side
length. We can observe that one side of the square, side 𝐴𝐷, is common with one side of the
right triangle. In fact, this side is the hypotenuse of the right triangle.
We can then recall that if we know the lengths of two sides of a right triangle, we
can calculate the length of the third side by applying the Pythagorean theorem. This states that in any right triangle, the sum of the squares of the two shorter
sides is equal to the square of the hypotenuse. If we use the letters 𝑎 and 𝑏 to represent the lengths of the two shorter sides and
𝑐 to represent the length of the hypotenuse, then this can be expressed as 𝑎
squared plus 𝑏 squared equals 𝑐 squared.
In triangle 𝐴𝐷𝐸 then, the Pythagorean theorem tells us that 𝐴𝐷 squared is equal
to 6.1 squared plus 6.4 squared. To solve for 𝐴𝐷, we can evaluate the squares and find their sum, giving 𝐴𝐷
squared equals 78.17.
Now we could continue and solve this equation for 𝐴𝐷 by square rooting, but let’s
think about what we’re trying to achieve here. The reason we want to find the length of 𝐴𝐷 is so we can calculate the area of
square 𝐴𝐵𝐶𝐷. But the area of a square is just equal to its side length squared. So if 𝐴𝐷 squared equals 78.17, then this is the area of the square and we’ve
already found what we needed. We can then complete the problem by evaluating the area of the triangle and
subtracting. The area of the triangle is 19.52 square units, and subtracting this from 78.17 gives
58.65.
We’re asked to give the answer to the nearest hundredth, but as this value is already
exact to two decimal places, there’s no need to round. So, we’ve found that the area of the shaded region is 58.65 square units.
