𝐴𝐵𝐶𝐷 is an isosceles trapezoid where 𝐴𝐵 equals 𝐴𝐷 equals 𝐷𝐶 equals eight centimetres and 𝐵𝐶 equals 16 centimetres. Find the area giving the answer to two decimal places.
So in the diagram, we have our trapezoid 𝐴𝐵𝐶𝐷 and just a quick word about the naming of this shape. In the US and Canada, you know this as a trapezoid. And in the rest of the world, this will be a trapezium. We’re told that there are three equal sides. 𝐴𝐵 𝐴𝐷, and 𝐷𝐶 are all eight centimetres. In a trapezoid, we will have a pair of parallel sides. Here, these will be 𝐴𝐷 and 𝐵𝐶. We can notice that our lengths 𝐷𝐶 and 𝐴𝐵 are the same, which is why we’re told that it is an isosceles trapezoid. An isosceles trapezoid has one pair of nonparallel sides congruent. To find the area of the trapezoid, we use the formula that the area of a trapezoid is equal to 𝐴 plus 𝐵 over two times ℎ, where 𝐴 and 𝐵 are the basis and ℎ is the perpendicular height.
Looking at our diagram, however, we can see that we’re not told the perpendicular height. But it will be possible to work this out if we know the base of the triangle and we use the Pythagorean theorem. To find the length of the base of this triangle, we can notice that, on the line 𝐵𝐶, we have an eight-centimetre segment. We know that the basis of the two triangles will be the same length, since we have an isosceles trapezoid. And we can call those both 𝑥. And we know that these three lengths will add up to 16 centimetres. Therefore, we must have four centimetres, eight centimetres, and four centimetres. If we take a closer look at one of the triangles in our trapezoid, we have a hypotenuse of eight centimetres and two other lengths.
We can use the Pythagorean theorem here which says that the square of the hypotenuse is equal to the sum of the squares on the other two sides. Substituting our values will give us eight squared equals eight squared plus four squared, where ℎ is, of course, the height of the triangle and not the hypotenuse. So 64 equals ℎ squared plus 16. To find ℎ squared by itself, we subtract 16 from both sides. Therefore, 48 is equal to ℎ squared. And finally, to find ℎ then, we take the square root of both sides, giving us that ℎ is equal to root 48. We leave our answer in this root form, since we’re not finished using this value of each in our calculations.
We can now work at the area of our trapezoid using the formula. Substituting our values, we have eight plus 16 over two times root 48. Eight plus 16 is 24. So we have 24 over two times root 48. And we could simplify our 24 over two as 12. So we have 12 times root 48. Notice, in the question that we were asked for value to two decimal places. So we can go ahead and use our calculator to simplify our answer. We get an answer of 83.138438 and so on square centimetres. And now to find the answer to two decimal places, we check the third decimal digit to see if it is five or more. And since it is, then our answer runs up to 83.14 square centimetres for the area of 𝐴𝐵𝐶𝐷.