Answer:
LF is 2 cm , LD is 4.24 (3√2) cm
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