Question

how to construct a square with diagonal 12cm​

Answer 1

Answer:

We first want to find the length of each side of the square so we can multiply its base by its height. Both numbers will be equal. A square with a diagonal of 12 cm will give us two right triangles, each with a hypotenuse of 12. The other legs (other sides) of both triangles will all be equal since a square, itself, has equal sides. From the unit circle we know (or at least I know) that any right triangle with non-hypotenuse sides having equal length will be proportionate with a right triangle with non-hypotenuse sides having length 1 and hypotenuse having length square root of 2. There may be several ways to find the length of the sides of the triangles that aren't the hypotenuse (same length of each side of the square) but I'd like to find that length by using cross multiplication. Substituting the length we don't know with the variable x, we’ll set x/12 equal to 1/root 2. These fractions correspond with the ratios of the respective triangles. From

x/12 = 1/root 2.

Cross multiply the first numerator with the second denominator and the second numerator with the first denominator. So x(root2)=12. Isolate x by dividing 12 by root 2. 12/root 2 is your new fraction but we don't want a radical in the denominator so multiplying the fraction by root 2/root 2 will give us 12(root2)/2, which can be simplified to 6 root 2. So 6 root 2 is the length of every non-hypotenuse side of the triangles and every side of the original square. Now that we have the length of a side we can deem the base and another we can deem the height, our area=base x height.

Area = (6 root 2) x (6 root 2) = 36(2) = 72

The area of a square whose diagonal is 12 cm is 72 squared centimeters.