Given: klmn is a parallelogram m∠n=3m∠k, lf ⊥ kn , ld ⊥ nm kf=2 cm, fn=4 cm find: lf, ld

LF is 2 cm , LD is 4.24 (3√2) cm

Step-by-step explanation:

* Lets look to the attached figure to solve the problem

- In parallelogram KLMN

∵ m∠N = 3 m∠K

∵ m∠N + m∠K = 180° ⇒ adjacent angle in parallelogram

- Replace the m∠N by 3 m∠K

∴ 3 m∠K + m∠K = 180° ⇒ add the like term

∴ 4 m∠K = 180° ⇒ divide both sides by 4

∴ m∠K = 45°

∵ LF ⊥ KN

∴ m∠LFK = 90°

∵ LD ⊥ NM

∴ m∠LDM = 90°

- In Δ LFK

∵ m∠LFK = 90° ⇒ proved

∵ m∠K = 45° ⇒ proved

∵ FK = 2 cm

- To find LF use the tangent function

∵ tan ∠K = opposite/adjacent

∵ The opposite is LF and the adjacent is FK

∴ tan (45) = LF/2

- Multiply both sides by 2

∴ LF = 2 tan (45) = 2 × 1 = 2

∴ LF is 2 cm

- Each two opposite sides in parallelogram are equal and each two

 opposite angles are equal

∴ m∠M = m∠K = 45° ⇒ opposite angles in parallelogram

∵ KF = 2 cm , FN = 4 cm

∵ KN = KF + FN

∴ KN = 2 + 4 = 6

∴ KN = 6 cm

∵ KN and ML are opposite sides in parallelogram

∴ ML = 6 cm

- In Δ LDM

∵ m∠LDM = 90° ⇒ proved

∵ m∠M = 45° ⇒ proved

∵ LM = 6 cm ⇒ proved

- To find LD use the sine function

∵ sin ∠M = opposite/hypotenuse

∵ The opposite is LD and the hypotenuse is LM

∴ sin (45) = LD/6

- Multiply both sides by 6

∴ LD = 6 sin (45) = 6 × √2/2 = 3√2 ≅ 4.24

∴ LD is 4.24 (3√2) cm

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