ABCD is a rectangle of dimensions 12 cm and 5 cm.AEFC is a rectangle drawn in such a way that thediagonal AC of the first rectangle is one of its sidesand side opposite to it is touching the first rectangleat D as shown in figure. What is the ratio of the areaof rectangle ABCD to AEFC?
Question
ABCD is a rectangle of dimensions 12 cm and 5 cm.AEFC is a rectangle drawn in such a way that thediagonal AC of the first rectangle is one of its sidesand side opposite to it is touching the first rectangleat D as shown in figure. What is the ratio of the areaof rectangle ABCD to AEFC?
Solution
To find the ratio of the area of rectangle ABCD to AEFC, we need to calculate the areas of both rectangles.
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Calculate the area of rectangle ABCD: The dimensions of ABCD are given as 12 cm and 5 cm. Area of ABCD = length * width = 12 cm * 5 cm = 60 cm²
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Calculate the area of rectangle AEFC: AEFC is a rectangle drawn in such a way that the diagonal AC of ABCD is one of its sides and the side opposite to it is touching ABCD at point D. Since AC is the diagonal of ABCD, we can use the Pythagorean theorem to find its length. AC² = AB² + BC² AC² = 12 cm² + 5 cm² AC² = 144 cm² + 25 cm² AC² = 169 cm² AC = √169 cm AC = 13 cm
The side opposite to AC in AEFC is touching ABCD at point D, so it has the same length as BC, which is 5 cm.
Area of AEFC = length * width = AC * BC = 13 cm * 5 cm = 65 cm²
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Calculate the ratio of the areas: Ratio of the area of ABCD to AEFC = Area of ABCD / Area of AEFC Ratio = 60 cm² / 65 cm²
Therefore, the ratio of the area of rectangle ABCD to AEFC is 60 cm² : 65 cm², which can be simplified to 12 : 13.
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