a What can you say about the diagonals in a square?
B
b Use the Pythagorean Theorem to evaluate the length of the diagonals in each square.
A
a See solution.
B
b 16+8sqrt(2) in.
Practice makes perfect
a Let's begin with recalling that diagonals in a square bisect opposite angles. Since diagonals of the square contain both legs of the trapezoids, we can state that the base angles of these trapezoids have the same measure, 45^(∘). Moreover, the bases of these trapezoids are parallel. With this information, the trapezoids are isosceles.
b We will start with recalling that the perimeter of a square is its side length multiplied by 4, as all sides of this figure are congruent. Therefore, we can evaluate the side lengths of both squares by dividing their perimeters by 4.
Side Length of a Tile:& 48/4=12 in.
Side Length of the Red Square:& 16/4=4 in.
Let's look at the picture and add the information we found.
To evaluate the perimeter of one of the trapezoids, we need to find the measures of the legs of the trapezoid. To do this, we can use the Pythagorean Theorem, as all angles in a square are right angles.
First, let's evaluate the length of the diagonal of the tile. Recall that, according to the Pythagorean Theorem, the sum of squared legs of a right triangle is equal to its squared hypotenuse, which is the diagonal in this case. Let's call it c.
Notice that the difference between the diagonal of the tile and the diagonal of the red square is equal to two leg lengths of one trapezoid. If we call the leg length x, we can evaluate this length by substituting c and d.
Finally, as we know all side lengths of the trapezoid, we can evaluate its perimeter. Remember that we determined in the previous part that these trapezoids are isosceles so their legs are congruent.
Let's add all four sides lengths of the trapezoid.
12+ 4+ 4sqrt(2)+ 4sqrt(2)=16+8sqrt(2)
The perimeter of each trapezoid is 16+8sqrt(2).
