The diagonal AC of the ABCD rectangle is 3 cm and makes an angle of 37 degrees with the AD side

| July 13, 2021 | education

The diagonal AC of the ABCD rectangle is 3 cm and makes an angle of 37 degrees with the AD side. Find the area of the rectangle ABCD.

We know the length of the AC diagonal, which is 3cm.

We also know the CAD angle of 37 degrees. Let’s draw a rectangle.

In order to solve this problem, we need to find the lengths of the two sides AD and CD. Let’s find them through the cosine function and the Pythagorean theorem.

The cosine in a right-angled triangle is the ratio of the adjacent side to the hypotenuse cosCAD = AD / AC. From this formula, we can derive the formula for finding the side AD = cosCAD * AC. It remains only to add values, the value of cos37 can be found in the Bradis table, it is equal to 0.79, rounded to 0.8.

So AD = 0.8 * 3 = 2.4cm.

We find the second side of CD through the Pythagorean theorem, which says: the square of the hypotenuse is equal to the sum of the squares of the legs. I.e:

AC ^ 2 = AD ^ 2 + CD ^ 2 we derive the formula CD ^ 2 for this we transfer the value of AD ^ 2 through the equal sign, changing the sign.

CD ^ 2 = AC ^ 2 – AD ^ 2.

Substitute the value CD ^ 2 = 3 ^ 2 – 2.4 ^ 2 = 3.24cm.

We extract the root CD = 1.8 cm.

Now we can find the area of ​​a rectangle, which is equal to the product of its sides: SABCD = AD * CD.

SABCD = 2.4 * 1.8 = 4.32cm2.

Answer: SABCD = 4.32cm2.



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