The diagram shows a 1 by x rectangular plank which fits neatly inside a 10 ×10 square frame. What is the value of x?
Note there are 2 big 45-45-90triangles, 2 small triangles 45-45-90 triangles and a rectangle in the middle.
Thus, the sum of all these areas is the area of big square which is 10x10=100.
The area of the rectange in the middle is 1xX=X
The area of the small triangles can easily be determined because they add up to a perfect square with diagnol of 1, thus the area is 1/2
The area of the big triangles can easily be determined because they add up to a perfect square with diagnol of x thus thee area is (1/2)x^2
Thus we have,
Thus,
Thus,
Thus,
Thanks . I had a question though, do you find those little triangles by going through this reasoning.
We know the diagonals of these triangles is of 1 unit long then their sides are of lengths 1 / SQRT 2.
Therefore one little triangle on its own has area 1/4 but both have area 1/2.
You can do that.Originally Posted by Natasha1
I actually used a different reasoning. Since as I said the little triangles combine to give you a square. With a diagnol of 1 I used the following theorem. If the diagnols of a rhombus are d1 and d2 then is area is (1/2)d1d2 . Since a square is a rhombus and both its diagnols are equal we have its area as,
Hello, Natasha!
1) The diagram shows a 1 by x rectangular plank which fits neatly inside a 10 ×10 square frame.
What is the value of x?We see that is a isosceles right triangle with hypotenuseCode:A D B * - * - - - - * | *1 * | G * * x | | * * | | * * | | * * E | * * | * - - - - * - * D F C
. . Then:
Since is an isosceles right triangle with legs
. . then: