Quadrilaterals · Lesson 3 of 13
Squares: Special Rectangles
“See how a square extends rectangle properties with equal sides, perpendicular diagonals and angle bisection.”
• Explain why every square is a rectangle. • Prove that square diagonals are perpendicular. • Find the angles formed by a square’s diagonals. • Construct a square from a given diagonal.
Suppose you shorten the long sides of a rectangular picture frame until they match its short sides, keeping every corner a right angle. You now have a square. Have you left the rectangle family? To decide, return to the definition instead of the familiar picture of a long rectangle. The square still satisfies every rectangle condition, but it has an additional equality that gives its diagonals extra properties.
This lesson connects the two names and returns to the wooden-strip construction. We will find the extra condition that changes a rectangular frame into a square.
A Special Rectangle
A square has four right angles and four equal sides. Since its four angles are right angles, it is a rectangle. The reverse is not always true: a rectangle with length 6 cm and width 3 cm does not have four equal sides.
Think of the set of squares as a smaller region inside the set of rectangles. Belonging to the smaller region automatically means belonging to the larger one. One figure can correctly have both names.
A quadrilateral in which all four angles are 90° and all four sides have the same length.
Problem
One frame is 5 cm by 5 cm with right angles. Another is 5 cm by 2 cm with right angles. Name each.
Deduction 5 — What should be the angle formed by the diagonals?
Let diagonals AC and BD of square ABCD meet at O. Because a square is a rectangle, they are equal and bisect each other. Compare triangles BOA and BOC: BA = BC, OA = OC, and BO is common.
The triangles are congruent by SSS. Thus ∠BOA = ∠BOC. These angles form a straight angle along AC, so their sum is 180°. Two equal angles adding to 180° must each be 90°.
Conversely, draw equal diagonals that bisect each other at 90°. The four small triangles each have two equal half-diagonal lengths and the included right angle. They are congruent by SAS, making all four outer sides equal. The earlier rectangle result supplies four right angles, so the outer figure is a square.
Problem
Construct a square with diagonal AC = 8 cm.
Properties of a Square
A square inherits the equal, parallel opposite sides and equal, bisecting diagonals of a rectangle. It adds four equal sides and perpendicular diagonals. These facts are connected; they are not separate accidents in a drawing.
Draw diagonal AC. Triangle ADC has AD = DC and a right angle at D. Its two remaining angles are equal and sum to 90°, so each is 45°. The same reasoning in triangle ABC shows that AC splits both corners it reaches into equal halves. Repeating with BD gives the result at the other two corners.
Problem
AM and PL are perpendicular diameters of a circle with centre O. What is the boundary APML?
Figure it Out
Quiz
Which statement is always true?
The diagonals of a square meet at:
A diagonal of a square divides a corner angle into:
Which extra side condition turns a rectangle into a square?
A square's diagonals are:
Practice Problems
- A square has diagonal 14 cm. Find the distance from the centre to each vertex.
- Find the angles at a corner cut by a square’s diagonal.
- Construct a square with diagonal 6 cm without a protractor.
- Why is a rectangle with sides 8 cm and 3 cm not a square?
- Two equal line segments cross at 90°, but do not bisect each other. Is a square guaranteed?
1. The equal diagonals are bisected at their common centre. 2. Each half-diagonal is 14 ÷ 2 = 7 cm, so every vertex is 7 cm from the centre.
1. Each corner is 90° and the diagonal bisects it. 2. Both parts are 90° ÷ 2 = 45°.
1. Draw AC = 6 cm. With equal compass radii greater than 3 cm, draw arcs from A and C that meet above and below AC. Join their intersections to make its perpendicular bisector. 2. Call the midpoint O. Mark B and D on this line on opposite sides, each 3 cm from O. 3. Join the endpoints in order. Equal, bisecting, perpendicular diagonals guarantee the square.
1. It satisfies the right-angle condition. 2. But its adjacent sides are unequal, so it fails the four-equal-sides condition for a square.
1. A square needs all three diagonal conditions: equal lengths, perpendicularity and mutual bisection. 2. Here the midpoint condition is missing. 3. Thus the information does not guarantee a square.
Key Takeaways
• Every square is a rectangle, but a rectangle need not be a square. • A square has four equal sides and four right angles. • Its diagonals are equal and bisect each other at 90°. • Each diagonal bisects its vertex angles into 45° parts. • All three diagonal conditions matter when constructing a square.