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Kite ABCD is such that AB=BC=40 cm
and AD = CD = 74 cm. If AC = 48 cm,
Calculate: (a) length BD, (b) A<DC.
​

Sagot :

sqdancefan

Answer:

  (a)  BD = 102 cm

  (b) ∠ADC = 37.8°

Step-by-step explanation:

Given kite ABCD with AB = BC = 40 cm, AD = CD = 74 cm, AC = 48 cm, you want the length of BD and the measure of angle D.

Diagonals

The diagonals of a kite cross at right angles. If we call the point of intersection E, then the Pythagorean theorem tells us ...

  AE² +BE² = AB²

  AE² +DE² = AD²

Using these relations, we can find BE and DE.

  BE = √(AB² -AE²) = √(40² -24²) = 32

  DE = √(AD² -DE²) = √(74² -24²) = 70

The length of diagonal BD is ...

  BD = BE +DE = 32 +70

  BD = 102 . . . . cm

Angle

DE bisects angle D. The measure of the angle can be found from the cosine relation:

  cos(D/2) = DE/AD = 70/74

  ∠D = 2·arccos(70/74)

  ∠D ≈ 37.8493°

The measure of angle ADC is about 37.8°.

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