Question

In the rhombus ABCD ∠B = 60°, AB = 3x – 7, BC = x + 5, AC = y + 3, x, y are = ______.

विकल्प

• x = 5, y = 10

• x = 6, y = 8

• x = 3, y = 5

• x = 5, y = 7

Answer 1

In the rhombus ABCD ∠B = 60°, AB = 3x – 7, BC = x + 5, AC = y + 3, x, y are = x = 6, y = 8.

Explanation:

We are given a rhombus ABCD with:

• ∠B = 60°
• AB = 3x – 7
• BC = x + 5
• AC = y + 3

We need to find the values of x and y.

Step 1: Use property of rhombus all sides are equal

AB = BC

⇒ 3x – 7 = x + 5

Solve for x:

3x – x = 5 + 7

2x = 12 ⇒ x = 6

Step 2: Use diagonals and angle

In a rhombus:

• Diagonals bisect each other at 90°
• Each diagonal bisects the angles at the vertices

Given ∠B = 60° and diagonals intersect at 90°, triangle ΔABC is formed with:

• ∠B = 60°
• Side AB = BC (rhombus)

So triangle ABC is isosceles and using Law of Cosines or geometrically, we determine AC, the diagonal.

Since we’re told AC = y + 3 and with x = 6 ⇒ AB = 3(6) – 7 = 11

So triangle ABC has:

• Two sides = 11
• Included angle = 60°

Use cosine rule:

AC² = AB² + BC² – 2(AB)(BC) cos(60°)

AC² = 11² + 11² – 2(11)(11)(0.5)

= 121 + 121 – 121

= 121

⇒ AC = `sqrt(121)` = 11

Now, y + 3 = 11

⇒ y = 8