The carpenter's strips
Do the two diagonals of a rectangle match?
A frame from two strips
A carpenter wants to make a rectangle from two thin strips of wood. The strips will be the diagonals: each joins a corner to the opposite corner. She has one strip, 8 cm long.
How long should the other strip be?
What this lesson covers
The idea
The diagonals of a rectangle always have the same length, as shown by congruent triangles (SAS).
A frame from two strips
A carpenter wants to make a rectangle from two thin strips of wood. The strips will be the diagonals: each joins a corner to the opposite corner. She has one strip, 8 cm long.
How long should the other strip be?
- Shorter than 8 cm
- Exactly 8 cm
- Longer than 8 cm
- It depends on the rectangle
Two strips, one rectangle
Drag the corner C to change the rectangle. The two bars show the lengths of the strips AC and BD. Then flip a copy of AC over to see if it fits BD.
The strips always match
The diagonals of a rectangle always have the same length, as shown by congruent triangles (SAS).
In triangles ADC and DAB: AB = CD (opposite sides), ∠A = ∠D = 90°, and AD is common. So they are congruent by SAS, and AC = BD.
So the carpenter's other strip must also be 8 cm long.
Notes
The diagonals of a rectangle always have the same length, as shown by congruent triangles (SAS).
Check yourself
A carpenter makes a rectangle frame. One diagonal strip is 13 cm long. How long is the other diagonal strip?
Answer: 13 cm
The diagonals of a rectangle are equal. The other strip is also 13 cm.
In rectangle ABCD, AC = 3x + 1 cm and BD = 25 cm. Find x.
Answer: 8
AC = BD, so 3x + 1 = 25. Then 3x = 24 and x = 8.
Which pair of triangles shows why AC = BD in rectangle ABCD?
A quadrilateral has diagonals of 9 cm and 7 cm. Can it be a rectangle?
- ∆AOB and ∆COD. Those two give AO = CO and BO = DO. To get AC = BD we use triangles that contain the whole diagonals.
- ∆ADC and ∆DAB: AD is common, AB = CD, and both angles at A and D are 90° — correct. Yes! SAS makes ∆ADC ≅ ∆DAB, so AC = BD.
- ∆ABC and ∆ADC. Both share AC. They do not compare AC with BD.
- Yes, any quadrilateral can be. The diagonals of a rectangle must be equal.
- No, the diagonals of a rectangle are equal — correct. Yes! 9 cm and 7 cm are not equal, so it cannot be a rectangle.
- Yes, if its angles are 90°. If its angles were all 90° it would be a rectangle, and then the diagonals would be equal. These are not, so it cannot have four 90° angles.