Rectangles and Rhombuses · Little Blue Book, Visualized

Try this

A rectangle is a parallelogram with one right angle. A rhombus is a parallelogram with one pair of equal neighboring sides.

Click “Snap to: Rectangle”. Look at the diagonal readings: the two diagonals are now equal. Drag $C$ a little to break the right angle, and “equal” disappears at once.
Click “Rhombus”: the four sides are equal and the diagonals are perpendicular, but usually not equal.
For the rectangle, turn on “Angles”: all four angles are $9 0^{\circ}$ . For the rhombus, opposite angles are equal but usually not right angles; turn on “Side lengths” too, and all four sides are the same length.
Click “Square”: the rectangle and rhombus labels are both lit, and the diagonals are both equal and perpendicular.

Predict first

The two diagonals of a quadrilateral are equal. Must it be a rectangle?

Yes, alwaysNot necessarilyNo, never

Choose an answer first; then you can see the result.

Why: it is all in the diagonals

From parallelogram to rectangle, rhombus and square

Step 1 / 4

Start: a parallelogram
The diagonals bisect each other, and that is all. The distances from $O$ to the four vertices are equal in pairs: $O A = O C$ , $O B = O D$ .
Add a right angle → rectangle
Equal neighboring sides instead → rhombus
Both at once → square

Two handy consequences:

The diagonals of a rectangle are equal and bisect each other, so the median to the hypotenuse of a right triangle is half the hypotenuse (complete the right triangle to a rectangle).
The two perpendicular diagonals cut a rhombus into four congruent right triangles, so the area of a rhombus $= \frac{1}{2} \times$ the product of its diagonals.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: Equal diagonals ⇒ $A B C D$ is a rectangle

$D$ is locked, so $A B C D$ is always a parallelogram. Drag $C$ until the diagonals light up as “equal”: at that moment the “rectangle” label lights up too, every time.

Practice

Exercise 1Basic

A rectangle has a side of length $6$ and a diagonal of length $10$ . Find its area.

Exercise 2Basic

The diagonals of a rhombus are $6$ and $8$ . Find its perimeter.

Exercise 3Advanced

The diagonals of rectangle $A B C D$ meet at $O$ , $∠ A O B = 6 0^{\circ}$ , and $A B = 4$ . Find the length of the diagonal $A C$ .

Exercise 4Challenge

In rectangle $A B C D$ , $A B = 6$ and $B C = 8$ , and the diagonals meet at $O$ . $P$ is any point on side $B C$ , with $P E ⊥ A C$ at $E$ and $P F ⊥ B D$ at $F$ . Find $P E + P F$ .

Mistake Museum · Exhibits from this stop

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Exhibit

A quadrilateral with equal diagonals is a rectangle

In quadrilateral

A B C D

A C = B D

, so it is a rectangle.

What went wrong? Think first, then open

Diagnosis The criterion says “a parallelogram with equal diagonals is a rectangle”. Drop “parallelogram” and an isosceles trapezoid, or even a very irregular quadrilateral, can have equal diagonals.

First prove it is a parallelogram (for example, the diagonals bisect each other), then add equal diagonals. Together: a quadrilateral whose diagonals are equal and bisect each other is a rectangle.

“The criteria for rectangles and rhombuses all stand on the shoulders of “parallelogram””

Exhibit

Using “half the product of the diagonals” for any quadrilateral

A parallelogram has diagonals

6

and

8

, so its area is

\frac{1}{2} \times 6 \times 8 = 24

What went wrong? Think first, then open

Diagnosis $S = \frac{1}{2} A C \cdot B D$ needs perpendicular diagonals (rhombus, square, kite). In a general parallelogram the diagonals do not meet at a right angle, and the area is smaller than this.

Only use this formula once you know the diagonals are perpendicular. Otherwise use “base × height”.

“A formula comes with its conditions: perpendicular first, then ½·d₁·d₂”