ABCD is a rectangle of dimension 12cm and 5cm.AEFC is a rectangle drawn in such a way that the diagonal AC of the first rectangle is one of its sides and side opposite to it is touching the first rectangle at D as shown in the figure.What is the ratio of the area of rectangle ABCD to AEFC?
Answers
The ratio of the area of rectangle ABCD to AEFC is 1:1.
Step-by-step explanation:
Referring to the figure attached below,
ABCD is given a rectangle with dimensions
AB = CD= 12 cm and AD = BC = 5 cm ….. [opposite sides of rectangle are equal in length] …. (i)
Also, AEFC is another rectangle drawn in such a way that diagonal AC is its length and its opposite side touched point D of the rectangle ABCD.
Step 1:
Let’s consider ∆ABC, using Pythagoras theorem, we get
AC² = AB² + BC²
⇒ AC = √[12² + 5²]
⇒ AC = √[169]
⇒ AC = 13 cm
Since AC = 13 cm is one of the lengths of rectangle AEFC, therefore its opposite facing side EF will also be 13 cm.
Step 2:
Let’s take ED as “x” cm then DF will be “(13 – x)” cm.
Consider ∆AED, using Pythagoras theorem, we get
AD² = AE² + ED²
⇒ 5² = AE² + x² ….. [substituting value of AD from (i)]
⇒ AE² = [25 - x²] …… (ii)
Similarly, consider ∆CFD, using Pythagoras theorem, we get
CD² = CF² + FD²
⇒ 12² = CF² + (13 – x)² ….. [substituting value of AB from (i)]
⇒ 144 = CF² + 169 – 26x + x²
⇒CF² = [-25 + 26x – x²] …… (iii)
Since AE = CF (opposite sides of rectangle AEFC), so we will equate eq. (ii) & (iii),
[25 - x²] = [-25 + 26x – x²]
⇒ 50 = 26x
⇒ x = 50/26
⇒ x = 1.92 cm
Substituting the value of x in eq. (ii), we get
AE² = [25 - x²]
⇒ AE = [25 – (1.92)²]
⇒ AE = [21.31]
⇒ AE = 4.61 cm
∴ AE = CF = 4.61 cm
Step 3:
Thus,
The ratio of the area of rectangle ABCD to AEFC is given by,
= [AB * BC] / [AC * AE]
= [12 * 5] / [13 * 4.61]
= [60] / [59.93]
= [60] / [60] ….. [taking approximate value of 59.93 as 60]
= 1 : 1
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