Kites · Years 8–9
In Kites (Years 8–9) you practise 3 skills: Angles in a kite · The diagonals of a kite · A kite on a coordinate grid. A kite is a quadrilateral with two pairs of equal adjacent sides. The diagonal joining the two vertices between the equal sides is the main diagonal, and it is the kite's line of symmetry: it bisects the two angles at its ends, and the other two angles are equal. The angles of a quadrilateral add up to 360°, so when you know the two angles on the main diagonal, each of the other two is half of what is left of 360°.
The diagonals of a kite are perpendicular, and the main diagonal bisects the other one. So they split the kite into four right triangles, and Pythagoras' theorem works in them: half the shorter diagonal and a part of the main diagonal give a side. The area of a kite is the product of its diagonals divided by 2. On a coordinate grid, when the main diagonal lies on the vertical axis, the fourth vertex is the reflection of the opposite vertex in that axis. To check whether a quadrilateral is a kite, compare the adjacent sides: work out their squared lengths and see whether they are equal.
Tip: Draw the main diagonal first: it splits the kite into two congruent triangles, and whatever is true on one side is true on the other.
What you learn
Angles in a kite
For example: In the kite ABCD the adjacent sides are equal: AB = AD, CB = CD . Given ∢B = 75°, ∢C = 160°, ∢D = 75° . Find ∢A .
Answer and solution
The answer: 50 °
The side angles are equal, so: ∢A = 360° − 75° − 160° − 75° = 50°
The diagonals of a kite
For example: In the kite ABCD the adjacent sides are equal: AB = AD, CB = CD . The diagonals meet at O , and BD = 6 cm. Find BO .
Answer and solution
The answer: 3 cm
The main diagonal bisects BD : BO = 6 ÷ 2 = 3
A kite on a coordinate grid
For example:
The points A(0, 2), B(4, −6), C(0, −8) are vertices of the kite ABCD . Its main diagonal AC lies on the vertical axis. What are the coordinates of D ?Answer and solution
The answer: (−4, −6)
A reflection in the vertical axis: the first coordinate changes sign and the second stays. D(−4, −6)
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).