Diagonals of a kite
The diagonal BD of a kite is like a mirror line.
What does BD do?
In kite ABCD, AB = BC and CD = DA. The diagonal BD joins the two corners where the equal sides meet. The other diagonal is AC.
What do you think the diagonal BD does?
What this lesson covers
The idea
In a kite ABCD with AB = BC and CD = DA, the diagonal BD bisects ∠ABC and ∠ADC, and bisects the diagonal AC at right angles.
What does BD do?
In kite ABCD, AB = BC and CD = DA. The diagonal BD joins the two corners where the equal sides meet. The other diagonal is AC.
What do you think the diagonal BD does?
- It cuts the angles at B and D in half
- It cuts the other diagonal AC in half
- Both of those, and it meets AC at a right angle
Drag the corner
The diagonal BD stays put. Drag A: C follows, so the shape is always a kite. Watch the numbers. Which ones are always equal? What is ∠AOB?
BD is a mirror line
In a kite ABCD with AB = BC and CD = DA, the diagonal BD bisects ∠ABC and ∠ADC, and bisects the diagonal AC at right angles.
In every kite the two angles at B matched, the two angles at D matched, AO = OC, and ∠AOB was always 90°.
Why? Compare ∆ABD and ∆CBD: AB = CB, DA = DC, and BD is common. They are congruent by SSS. So ∠ABD = ∠CBD and ∠ADB = ∠CDB: BD bisects ∠ABC and ∠ADC.
Now compare ∆ABO and ∆CBO: AB = CB, ∠ABO = ∠CBO, and BO is common. They are congruent by SAS. So AO = CO, and ∠AOB = ∠COB.
Those two angles lie on a straight line and add up to 180°, so each is 90°. BD bisects AC at right angles.
Only BD does this. In general the two parts of BD are different, so AC does not cut BD in half.
Notes
In a kite ABCD with AB = BC and CD = DA, the diagonal BD bisects ∠ABC and ∠ADC, and bisects the diagonal AC at right angles.
Check yourself
In kite ABCD, AB = BC and CD = DA, and ∠ABC = 80°. How large is ∠ABD?
Answer: 40 °
BD bisects ∠ABC, so ∠ABD = 80° ÷ 2 = 40°.
The diagonals of kite ABCD meet at O, and AC = 12 cm. How long is AO?
Answer: 6 cm
BD bisects AC, so AO = OC = 12 cm ÷ 2 = 6 cm.
In kite ABCD the diagonals meet at O, and ∠OAB = 50°. How large is ∠ABO?
Answer: 40 °
BD meets AC at 90°, so ∠AOB = 90°. In triangle ABO: 50° + 90° + ∠ABO = 180°, so ∠ABO = 40°.
In this kite, AB = BC = 5 cm, CD = DA = 9 cm and BD = 10 cm. The diagonals meet at O. Which is true?
- AO = OC: BD cuts AC in half — correct. Yes! In every kite the diagonal BD bisects AC.
- BO = OD: AC cuts BD in half. Look at the picture: O is much closer to B than to D. In a kite, BD cuts AC in half, but AC does not cut BD in half.
- Both diagonals cut each other in half, as in a parallelogram. That is a parallelogram. In this kite only BD cuts the other diagonal in half.