Question

If JKLM is a rhombus, N is the midpoint of the diagonals, MK = 30,NL = 13 and m∡MKL = 41°. Find each measure.

What is the length of KL (rounded to the nearest tenth)

Answer 1

Answer:

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A Rhombus j k l m, Diagonals m k and n l intersect each other at point n.

Properties of Rhombus

1. All sides are equal.

2. Diagonals bisect each other at right angles.

3. Opposite sides are Parallel.

∠m k l= 41°

m k=30 cm

m n= n k=15 cm→→Diagonals bisect each other

n l = n j = 13 cm →→→Diagonals bisect each other

In right Δ m n l,

∠ m n l= 90°

By Pythagoras Theorem

(m l)²=(m n)²+(n l)²

(m l)²= (15)²+(13)²

(ml)²=225 + 169

(ml)²=394

m l =√394

m l=19.84 cm

So, m l=j k =k l= 19.8 4 cm

Also,In Δ n k l

90°+41° +∠JLK=180°

∠j l k=180°-131°

∠ j l k=49°

In Δn j k and Δ n k l

n j= n l→→Diagonals bisect each other at right angles.

∠j n k=∠k n l=90°→→Diagonals bisect each other at right angles.

side n k is common.

Δn j k ≅ Δ n k l →→[SAS]

∠n k l = ∠ n j k→→→→[CPCT]

∠n l k =∠ n k j→→→→CPCT

As, opposite sides of rhombus are parallel, that is , m l║ j k and m j║l k, and m k and j l are Transversal.So alternate interior angles are equal.

∠m k j=∠ k m l=49°

∠j m k=∠m k l=41°

∠m j l=∠k l j=49°

∠ m l j=∠k j l =41°

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