# What is the length of a diagonal of a square if its area is 98 square feet?

Aug 6, 2018

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Length of the diagonal is color(blue)(14 feet (approximately)

#### Explanation:

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Given:

A square $A B C D$ with area of color(red)(98 square feet. What do we need to find?

We need to find the length of the diagonal.

Properties of a Square:

1. All the magnitudes of sides of a square are congruent.

2. All the four internal angles are congruent, angle = ${90}^{\circ}$

3. When we draw a diagonal, as is shown below, we will have a right triangle, with the diagonal being the hypotenuse. Observe that $B A C$ is a right triangle, with the diagonal $B C$ being the hypotenuse of the right triangle.

color(green)("Step 1":

We are given the area of the square.

We can find the side of the square, using the area formula.

Area of a square: color(blue)("Area = " "(Side)"^2

rArr "(Side)^2=98

Since all the sides have equal magnitudes, we can consider any one side for the calculation.

$\Rightarrow {\left(A B\right)}^{2} = 98$

$\Rightarrow A B = \sqrt{98}$

$\Rightarrow A B \approx 9.899494937$

$\Rightarrow A B \approx 9.9$ units.

Since all the sides are equal,

$A B = B D = C D = A D$

Hence, we observe that

$A B \approx 9.9 \mathmr{and} A C = 9.9$ units

color(green)("Step 2":

Consider the right triangle $B A C$

Pythagoras Theorem:

${\left(B C\right)}^{2} = {\left(A C\right)}^{2} + {\left(A B\right)}^{2}$

${\left(B C\right)}^{2} = {9.9}^{2} + {9.9}^{2}$

Using the calculator,

${\left(B C\right)}^{2} = 98.01 + 98.01$

${\left(B C\right)}^{2} = 196.02$

BC=sqrt(196.02

$B C \approx 14.00071427$

$B C \approx 14.0$

Hence,

the length of the diagonal (BC) is approximately equal to color(red)(14 " feet."

Hope it helps.

Aug 6, 2018

14

#### Explanation:

The side is the square root of the area

$S \times S = A$

S = $\sqrt{98}$

The diagonal is the hypotheus of a right triangle formed by the two sides so

${C}^{2} = {A}^{2} + {B}^{2}$

Where C = the diagonal A = $\sqrt{98}$ , B = $\sqrt{98}$

so ${C}^{2} = {\left(\sqrt{98}\right)}^{2} + {\left(\sqrt{98}\right)}^{2}$

this gives

${C}^{2} = 98 + 98$ or

${C}^{2} = 196$

${\sqrt{C}}^{2} = \sqrt{196}$

$C = 14$

The diagonal is 14