Video Transcript
𝐴𝐵𝐶𝐷 is a quadrilateral, where
the measure of the angle 𝐵 is 90 degrees, the side 𝐴𝐵 has length eight
centimeters, the side 𝐵𝐶 is six centimeters, the side 𝐴𝐷 is 39 centimeters, and
the side 𝐶𝐷 is 30 centimeters. Find the area of 𝐴𝐵𝐶𝐷, giving
the answer to the nearest hundredth.
We want to find the area of the
given quadrilateral. So first sketching the
quadrilateral, we see that we can split it into two triangles. And let’s call these triangles T
one and T two. We see that triangle T one is a
right triangle. And we’re going to use the
Pythagorean theorem to find the unknown side length 𝐴𝐶. This tells us that for a
right-angled triangle with sides 𝑎, 𝑏, and 𝑐, where 𝑐 is the hypotenuse, 𝑐
squared is equal to 𝑎 squared plus 𝑏 squared.
For the triangle T one then, we
have the length of the side 𝐴𝐶 squared is equal to eight squared plus six
squared. That is 64 plus 36, which is
100. And taking the positive square root
on both sides, this gives us 𝐴𝐶 is the square root of 100, which is 10. And our square root is positive
since lengths are always positive. So, the hypotenuse of our right
triangle is 10 centimeters.
Making a note of this, since we’ll
need this to find the area of triangle T two, we can use the formula to find the
area of a right triangle — that’s one over two times the base multiplied by the
height — to find the area of triangle T one. So, for T one, the base is eight
centimeters, and the height is six centimeters. The area of T one is one over two
multiplied by eight multiplied by six. And that’s 24 centimeters
squared. So making a note of this and making
some space, now let’s look at our triangle T two.
We have all three side lengths;
that’s 𝐴𝐶 is equal to 10, 𝐴𝐷 is 39, and 𝐶𝐷 is 30. And let’s call these sides
lowercase 𝑎, 𝑏, and 𝑐 for convenience. Since we have the length of all
three sides of the triangle, to find the triangle’s area, we can use Heron’s
formula. This tells us that for the triangle
with sides 𝑎, 𝑏, and 𝑐, the area is given by the square root of 𝑠 multiplied by
𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is equal
to 𝑎 plus 𝑏 plus 𝑐 over two. And that’s the semiperimeter of the
triangle; that’s half of the perimeter.
We begin by calculating 𝑠, and
that’s 10 plus 39 plus 30 over two. That is 79 over two, which is
39.5. And we can use this value in
Heron’s formula to find the value of the area of the triangle T two. This gives us the area of triangle
T two is the square root of 39.5, that’s 𝑠, multiplied by 39.5 minus 10, that’s 𝑠
minus 𝑎, multiplied by 39.5 minus 39, which is 𝑠 minus 𝑏, multiplied by 39.5
minus 30, which is 𝑠 minus 𝑐. Evaluating our parentheses, this
gives us the square root of 39.5 multiplied by 29.5 multiplied by 0.5 multiplied by
9.5. And typing the argument of our
square root into our calculators, this gives us 5534.9375. Taking the square root to five
decimal places, that’s 74.39716 centimeters squared.
And now making some space, remember
we’re trying to find the area of the quadrilateral 𝐴𝐵𝐶𝐷. And that’s the area of triangle T
one plus the area of triangle T two. The area of triangle T one is 24
centimeters squared. And the area of triangle T two is
74.39716 centimeters squared to five decimal places. And so, the area of our
quadrilateral 𝐴𝐵𝐶𝐷 to five decimal places is 98.39716 centimeters squared. We’re asked to find the area to the
nearest one hundredth; that’s to two decimal places. So, the area of the quadrilateral
𝐴𝐵𝐶𝐷 to the nearest one hundredth is 98.40 centimeters squared.
