Let's start by drawing the quadrilateral using the given vertices. Let's name the vertices with the consecutive letters.
First let's find the lengths of the segments of our quadrilateral. To do this we will draw four right triangles with hypotenuses that are the quadrilateral's sides. Then we will use the Pythagorean Theorem.
Now let's recall that a rectangle is a parallelogram with four right angles. This means that if our quadrilateral has four right angles it is a rectangle. Let's try to determine if this is true. We will start with angle next to vertex D.
To do this, let's focus on triangle ACD. Notice that we can count squares to find the length of side AC.
Now, if the following equation is true then by the converse of the Pythagorean Theorem, the angle next to D is right.
a^2+ b^2= c^2
Let's substitute the side lengths of â–ł ACD.
We end with a true statement, so the angle next to vertex D is a right angle. Also, we can see that â–ł ABC has the same side lengths, so the angle next to vertex B is also a right angle.
Next, there are two more angles that need to be right if our figure is a rectangle. To check if they are right we can use the same idea as before. We should start by finding the length of segment BD. Let's call it d. To do this, we will draw a right triangle with a hypotenuse of BD.
Now we can use the Pythagorean Theorem to evaluate d.
The segment BD has a length of 5, just like segment AC. We can see that segment BD divides our quadrilateral into two triangles of the same sides as triangle ACD. This means that these two triangles are also right by the converse of the Pythagorean Theorem.
The given quadrilateral has four right angles and opposite sides that are congruent. This means that it is a rectangle.
