Video Transcript
In a triangle 𝐴𝐵𝐶, point 𝐷 lies
on side 𝐵𝐶, the projection from 𝐴 to 𝐷 is perpendicular to side 𝐵𝐶, 𝐴𝐶
equals 37.8, 𝐴𝐷 equals 10.08, and 𝐴𝐵 equals 10.76. Find the length of side 𝐵𝐶 to the
nearest tenth and then determine whether triangle 𝐴𝐵𝐶 is a right triangle or
not.
Let’s begin by using the
information we’ve been given to sketch the triangle 𝐴𝐵𝐶. We know that side lengths 𝐴𝐶 and
𝐴𝐵 are 37.8 and 10.76, respectively, and that the projection 𝐴𝐷 is perpendicular
to 𝐵𝐶 and has length 10.08. We’re asked first to find the
length of side 𝐵𝐶, which we can do using the Pythagorean theorem for right-angled
triangles. This says that for any right-angled
triangle, the square of the hypotenuse equals the sum of the squares of the other
two sides.
Since 𝐴𝐷 is perpendicular to
𝐵𝐶, angles 𝐴𝐷𝐶 and 𝐴𝐷𝐵 are right angles. So we can find the lengths of sides
𝐷𝐶 and 𝐵𝐷 using the Pythagorean theorem on the two triangles 𝐴𝐷𝐶 and
𝐴𝐷𝐵. This is possible because we already
know two side lengths for both triangles. We can then sum our results to get
the length of side 𝐵𝐶.
So let’s begin with triangle
𝐴𝐷𝐶. Since the side opposite the right
angle is side 𝐴𝐶, which has length 37.8, by the Pythagorean theorem, we have 37.8
squared equals 10.08 squared plus 𝐷𝐶 squared. And now subtracting 10.08 squared
from both sides and evaluating the left-hand side, we have 𝐷𝐶 squared equal to
1327.2336.
Now, taking the positive square
root on both sides, positive since lengths are always positive, we have 𝐷𝐶 equals
36.43121 and so on. So now making some space and making
a note of this, we can follow the same process with triangle 𝐴𝐷𝐵 to find side
length 𝐵𝐷. In this case, 𝐴𝐵 is the
hypotenuse. So, by the Pythagorean theorem, we
have 10.76 squared equals 10.08 squared plus 𝐵𝐷 squared. And subtracting 10.08 squared from
both sides and evaluating then taking the square root on both sides, we have 𝐵𝐷
equals 3.76446 and so on.
So now making some space and making
a note of this, we can find side length 𝐵𝐶 by summing our two results. This gives us that 𝐵𝐶 equals 40.2
units long to one decimal place. That’s to the nearest tenth. So now we have the length of the
third side 𝐵𝐶 in triangle 𝐴𝐵𝐶. We can use the converse of the
Pythagorean theorem to determine whether triangle 𝐴𝐵𝐶 is a right triangle or
not. The converse of the Pythagorean
theorem tells us that if the square of the longest side of a triangle is equal to
the sum of the squares of the other two sides, then the triangle is a right
triangle.
We know that the longest side in
our triangle 𝐴𝐵𝐶 is side 𝐵𝐶, since we’ve just found its length to be equal to
40.2. And this is longer than either of
the other two sides 𝐴𝐵 or 𝐴𝐶.
So now to determine whether
triangle 𝐴𝐵𝐶 is a right triangle or not, we evaluate the square of 40.2 on the
left-hand side and the sum of the squares of 10.76 and 37.8 on the right-hand
side. If the two sides of our equation
are equal, then by the converse of the Pythagorean theorem, the triangle must be a
right triangle. However, our right-hand side
evaluates to 1544.6176, and that doesn’t equal 1616.04, which is 40.2 squared.
So, since the square of the longest
side of triangle 𝐴𝐵𝐶 does not equal the sum of the squares of the other two
sides, it is not a right triangle. Hence, by using the Pythagorean
theorem, we found side length 𝐵𝐶 to be equal to 40.2 units. And by the converse of the
Pythagorean theorem, we’re able to use this to show that triangle 𝐴𝐵𝐶 is not a
right triangle.
