Buksis-8

8.5

What are you going to learn? To describe the properties of a kite To define a kite To find the formula of the area and perimeter of a kite

Key Term:

Some of you may have played a kite or seen some people flying kites as in the right figure. A

Now, look at the form of a flying kite and also look at the angles, sides D and the length of its diagonals.

B

• kite

Quadrilateral ABCD on Figure 8.6 is a kite with sides AB , BC , CD , C Figure 8.6

AD , diagonals AC and BD .

Now, you will learn how to find the formula of the area of a kite.

Mini-Lab Work in Groups Materials: grid paper, scissors, and a ruler. A B

O

D

C

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

1. On a piece of squared paper, draw a triangle and cut along the triangle. (figure (i)). 2. 2. 3. 4. 5. 6. 7. 8.

Draw a line in the center in the width of a rectangle as in Figure (ii) above. Fold the rectangle based on dotted lines. (Figure (iii)) Draw in dotted lines as in figure (iv) above. Cut the fold based on the dotted lines, until they form a half kite as in Figure (v). Open the fold until they form a new quadrilateral as in figure (vi). The quadrilateral is a kite. Name the kite ABCD and the intersection as point O. (Figure (vii)) Based on this activity, observe which the sides, angles and diagonals are. What can you conclude about the properties of the kite? Explain.

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The properties of a kite are as follows. A

B

┐

D

E

C

1. Two pairs of the sides close to each other are equal, namely AB = AD and BC = DC. 2. One pair of backside angles is equal, that is ∠ABC = ∠ADC. 3. One of the diagonals bisects the kite, that is ΔABC = ΔADC or AC is the axis of symmetry. 4. Diagonals are perpendicular to each other and one of the diagonals bisects the other, that is, AC ⊥ BD and BE = ED.

Figure 8.7

On the basis of the properties of a kite above, you can describe a kite.

Kite

The Area of Kite

A quadrilateral with diagonals perpendicular to each other and one of the diagonals bisects the other.

In words: The area of kite equals a half of the product of its diagonals.

┐ d2 d1

In symbol: Suppose A is an area of a kite with its diagonals d1 dan d2; then 1 A = x d1 x d2 2

Andi made a kite with its diagonals 30 cm and 50 cm long. What is the area of the kite made by Andi?

330 / Student’s Book – Quadrilaterals

Solution: Given Question Solution

: d 1 = 30 and d 2 = 50 : The area of the kite : For example, the area of Andi’s kite is A cm2, then 1 A = × d1 ×d2 2 1 A = × 30 × 15 2 A = 225 So, the area of Andi’s kite is 225 cm2.

Mathematics for Junior High School Year 7 / 331

1. Let ABCD be a kite with BE = 15 units, measure of ∠BCA = 30° and ∠DAC = 50°. Fill in the blanks below. BD = ED =

A

B E

D

C

Measure of ∠BAD =

°

Measure of ∠DCA =

°

Measure of ∠BEA =

°

Measure of ∠AED =

°

Area of ΔABC = area of Δ Area of ΔADE = area of Δ 2. 130 ° 40°

? ?

What are the values of x and y?

3. What are the values of x and y?

y°

y° A (5x)°

D

3m

m 4. What is the area of Ekite3 ABCD? B

P 13 m

18 m

R

40° x°

8m

C

5. What is the area of kite PQRS if ∠PQR is a right angle? S

Q

70°

Say whether the following statements are true or false.

6. A kite can be formed by an obtuse-angled triangles and its mirror on one side of the triangle. 7. A kite has two equal parallel sides. 8. A kite has an axis of symmetry. 9. The sum of four angles in kite is 360°. 10. The sum of opposite angles is 180°.

332 / Student’s Book – Quadrilaterals

11. Kite XYZW below has diagonals XZ and YW intersecting at point V. If XZ = 20 cm, YW = 30 cm, and VY = 7 cm, then XV= , VZ= , WV= , and measure of ∠YVZ=

°.

12. Determine the area of kite XYZW above. 13. Critical Thinking. Can two angles close to each other be supplementary? 14. A kite has its side by side lines of 9 cm and 12 cm in length. Calculate the perimeter. 15. Critical Thinking. Can two vertical angles of a kite be supplementary? 16. Show that the area of kite KLMN is 63 cm 2 , if LN = 12 cm, and KM = 10,5 cm. 17. Critical Thinking. There is a statement that the longest diagonal of a kite is called the symmetric line of that kite. Is it true? Explain it.

Mathematics for Junior High School Year 7 / 333

334 / Student’s Book – Quadrilaterals