For the given rhombus ABCD, find the value of x. Also find the length of the side of the rhombus, if AC = 8 cm and BD = 6 cm.
Answers
Properties of a Rhombus
Before finding the answer, let us revise the properties of a rhombus, which are going to help a lot in this question.
→ Each side of a rhombus is equal. Therefore, in the given rhombus, DA = AB = BC = CD.
→ Adjacent sides are supplementary, meaning the add up to 180°.
→ The diagonals bisect the angles of the rhombus. This means that in rhombus ABCD, ∠BAO = ∠DAO.
→ The diagonals bisect each other at right angles so, we have four right-angled triangles in the given rhombus: ΔDOC, ΔCOB, ΔBOA and ΔOAD.
Now, we're all set!
Answer
Given:
- ∠OAD = 56°
- AC = 8 cm
- BD = 6 cm
To find:
- the value of x
- the length of the side
Solution:
We know that ∠BAO = ∠DAO = 56° since the diagonals bisect the angles.
Therefore, ∠DAB = 56 + 56 = 112°
As ∠DAB and ∠ABC are adjacent, they are supplementary. So,
⇒ ∠DAB + ∠ABC = 180
⇒ 112 + ∠ABC = 180
⇒ ∠ABC = 180 - 112
⇒ ∠ABC = 68
If angle ABC = 68, then x = 68/2 (diagonals bisect the angles).
Therefore, the value of x is 34°.
Let us assume the triangle DOC. Here,
- DO = BD/2 = 6/2 = 3 cm
- CO = AC/2 = 8/2 = 4 cm
The hypotenuse of this triangle is the side of the rhombus.
According to Pythagoras Theorem, the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the sides. Therefore,
⇒ DO² + CO² = DC²
⇒ 3² + 4² = DC²
⇒ 9 + 16 = DC²
⇒ 25 = DC²
⇒ DC = √25
⇒ DC = 5 cm
Therefore, the length of the side of the rhombus is 5 cm.