Construct a rectangle in which one of the diagonals divides the opposite angles into 60° and 30°.
Answer
Start with a rough diagram of the rectangle ABCD, with the diagonal AC making 60° with AB and 30° with AD. Then decide the order of the steps: the length of AB is not given, so we may take it as we like.
Steps of construction
Step 1. Draw AB of any convenient length, say 4 cm. At B draw a perpendicular to AB.
Step 2. At A, draw a ray making an angle of 60° with AB. Where it cuts the perpendicular at B, mark the point C.
Step 3. At A draw a line perpendicular to AB. The point D must lie on this line.
Step 4, Method 1. At C draw a perpendicular to BC; it cuts the line through A at D. Method 2. Or open the compass to BC, put the tip at A and cut the perpendicular at A — that point is D, since AD = BC.
Join CD. ABCD is the required rectangle.
The diagonal AC splits the right angle at A into 60° and 30°. With AB = 4 cm, BC works out to about 6.9 cm.
Why the shape is fixed even though no length was given: the two angles fix how steep the diagonal is, and that fixes the ratio of the sides. Choosing AB = 4 cm simply decides the size — every such rectangle has the same shape, only bigger or smaller.
Q2.
Now ∠A is divided into two angles. One measures 60°. Check what the other angle is.
Answer
∠A = 90° (all angles of a rectangle are right angles)
one part = 60°
other part = 90 − 60 = 30° ✔
This matches the question, which asked for parts of 60° and 30°.
Why it must be 30°: the diagonal lies inside the corner, so the two parts together make up the whole corner. Since the whole corner is 90°, fixing one part at 60° automatically fixes the other at 30°. This is why the problem gives you two numbers that add up to 90 — if they did not, no rectangle could exist.
Check it yourself: measure the two parts of the opposite angle C as well. They are also 30° and 60°, but the other way round.
Q3.
Construct a rectangle where one of its sides is 5 cm and the length of a diagonal is 7 cm.
Answer
Rough diagram first: rectangle ABCD with DC = 5 cm and the diagonal DB = 7 cm. The base can be drawn at once; the trouble is to find B.
Steps of construction
Step 1. Draw the base DC = 5 cm.
Step 2. At C draw a perpendicular to DC and call this line l. The point B lies somewhere on l.
Step 3. Open the compass to 7 cm, place the tip at D and draw an arc cutting the line l. The cut is the point B (it is 7 cm from D and lies on l — both conditions at once).
Step 4. Draw a perpendicular to DC at D and a perpendicular to BC at B. They meet at A. (Or simply take the compass opening CB and cut from D.)
ABCD is the required rectangle. Check it against R1 and R2.
Side 5 cm and diagonal 7 cm. The orange arc of radius 7 cm, drawn from D, cuts the perpendicular l exactly at B. The second side comes out about 4.9 cm.
Why the arc finds B without trial and error: the arc contains all the points that are 7 cm from D, and the line l contains all the points that make a right angle at C. B has to satisfy both, so it must be where the two meet.
Q4.
Consider the point at which the circle and the line intersect. What is its distance from point D? If needed, check your figure. What do you observe?
Answer
Its distance from D is exactly 7 cm — the radius of the circle.
B is on the circle → DB = radius = 7 cm
B is on the line l → ∠BCD = 90°
What we observe: the crossing point satisfies both conditions at the same time, so it is the corner B we were hunting for. Measuring the third side gives CB ≈ 4.9 cm.
Why this is such a useful idea: “7 cm from D” by itself allows infinitely many points (a whole circle); “on the line l” by itself also allows infinitely many. Putting the two together leaves just one point. Nearly every construction in this chapter works this way — draw the two families of points and take their meeting point.
Q5.
Method 2: To locate the point B, was it necessary to draw the entire circle?
Answer
No. Only the small piece of the circle that crosses the line l is of any use, so it is enough to draw a short arc there.
Open the compass to 7 cm and place the tip at D.
Swing the pencil only near the line l — draw a light arc that crosses it.
Mark the crossing as B.
Tip: arcs keep the figure clean and are quicker to draw. A full circle also adds a second, unwanted crossing on the other side of DC, which can confuse you.
Q6.
Check if ABCD is indeed a rectangle satisfying properties R1 and R2.
Answer
Measure the finished figure:
R1 DC = AB = 5 cm, CB = DA ≈ 4.9 cm → opposite sides equal ✔
R2 ∠A = ∠B = ∠C = ∠D = 90° ✔
Extra check: the other diagonal AC also measures 7 cm ✔
Both properties hold, so ABCD is a rectangle with a 5 cm side and a 7 cm diagonal, exactly as asked.
Check it yourself: 5 cm across and 4.9 cm up should give 7 cm along the slant. Measure the diagonal DB once more — if it is not 7 cm, the perpendicular at C was probably not exact.