Video Transcript
Triangle 𝐴𝐵𝐶 is right angled at
𝐵. Given that the measure of angle 𝐴
is 30 degrees, that 𝐴𝐵 is equal to 37.6 centimeters, and 𝑀 is the circle that
lies on 𝐴 and 𝐵 and is tangent to line segment 𝐴𝐶 at 𝐴, determine the area of
triangle 𝐴𝐵𝑀 to the nearest tenth of a square centimeter.
We’ve been given this whole
description without an image. Our first job here is going to be
to sketch an image to represent the information we’ve been given. So we start at the beginning. We have triangle 𝐴𝐵𝐶, which has
a right angle at 𝐵. And we know that the measure of
angle 𝐴 is 30 degrees. Line segment 𝐴𝐵 measures 37.6
centimeters. The next thing we know is that
there is some circle 𝑀 that lies on 𝐴 and 𝐵. That means that 𝐴𝐵 is a chord of
the circle.
But we especially know that the
line segment 𝐴𝐶 is a tangent to the circle, which means we can sketch something
like this. 𝑀 is the center of that
circle. And then we know that the points
𝐴, 𝐵, and 𝑀 make a triangle. From here, we’ll need to use our
knowledge of both circles and triangles to figure out this area. The area of triangle 𝐴𝐵𝑀 is what
we want to find. We know to find the area of a
triangle, we have one-half the height times the base. The base of this triangle is
37.6. It’s the line segment 𝐴𝐵. So we can go ahead and add that
value. The height of this triangle is the
perpendicular distance from point 𝑀 to the base. And that’s the missing value we
need to solve for.
But before we can do that, we’ll
need to fill in more information. Now, we know that line segment 𝐴𝐶
is tangent to this circle at point 𝐴. And that means it forms a right
angle. We already know that the angle
𝐵𝐴𝐶 is 30 degrees. And that will make this remaining
portion of the right angle 60 degrees because together they form a right angle,
which is 90 degrees. Now, if 𝑀𝐴𝐵 measures 60 degrees,
we know that 𝐴𝐵𝑀 will measure 60 degrees as well. We know this because they’re both
opposite radii of the circle, which means line segment 𝐴𝑀 is equal in length to
line segment 𝐵𝑀. It also means the triangle we’re
trying to find the area of is an equilateral triangle, 60 degrees on all three
angles.
But again, in order for us to find
the height of this triangle, we need to know this distance. To find that, let’s try and zoom in
on this portion of our figure. We have a right angle and a
60-degree angle, which means the remaining angle will be 30 degrees. If we call this point 𝑑, when we
go back to our original figure, we can say that 𝐴𝑑 is going to be equal in length
to 𝑑𝐵. This is because the line segment
𝑀𝑑, the height of this triangle, is a perpendicular bisector. And that means the segment 𝑑𝐵 is
half 37.6, which is 18.8 centimeters. Looking at our triangle here, we
know all the angles and we know one side length.
In addition to that, we can say
that this is a special right triangle, a 30-60-90 triangle. And we know that the ratio of side
lengths in a 30-60-90 triangle occur in the ratio one to square root of three to
two. In this case, the side length
opposite 30 degrees measures 18.8. And the side length opposite the
60-degree angle is the value we’re interested in, the height of this triangle. To go from one to the square root
of three, you multiply by the square root of three, which means the height of our
triangle will be equal to 18.8 times the square root of three, which becomes 32.5625
continuing. And that’s a measure of
centimeters.
If we round to three decimal
places, we get a height of 32.563 centimeters. So we take that value, and we plug
it in to our area formula. Then we enter this into our
calculator. And we get 612.1844. And this is area, so it is a
measure of centimeters squared. We’re looking for this rounded to
the nearest tenth of a square centimeter. We look to the right of that, and
we have an eight, which means we will round up to 612.2 centimeters squared. The area of triangle 𝐴𝐵𝑀 is
612.2 centimeters squared.
