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

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

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Answer 1 · 2018-08-06T02:09:14

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

Explanation:

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**Given: **

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

We need to find the length of the diagonal.

Properties of a Square:

All the magnitudes of sides of a square are congruent.
All the four internal angles are congruent, angle = $90^{\circ}$ $90^@$
When we draw a diagonal, as is shown below, we will have a right triangle, with the diagonal being the hypotenuse.

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Observe that $B A C$ $BAC$ is a right triangle, with the diagonal $B C$ $BC$ being the hypotenuse of the right triangle.

$Step 1 :$ $color(green)("Step 1":$

We are given the area of the square.

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

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

$\Rightarrow {(S i d e)}^{2} = 98$ $rArr "(Side)^2=98$

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

$\Rightarrow {(A B)}^{2} = 98$ $rArr (AB)^2=98$

$\Rightarrow A B = \sqrt{98}$ $rArr AB=sqrt(98)$

$\Rightarrow A B \approx 9.899494937$ $rArr AB~~9.899494937$

$\Rightarrow A B \approx 9.9$ $rArr AB~~9.9$ units.

Since all the sides are equal,

$A B = B D = C D = A D$ $AB=BD=CD=AD$

Hence, we observe that

$A B \approx 9.9 and A C = 9.9$ $AB~~9.9 and AC=9.9$ units

$Step 2 :$ $color(green)("Step 2":$

Consider the right triangle $B A C$ $BAC$

Pythagoras Theorem:

${(B C)}^{2} = {(A C)}^{2} + {(A B)}^{2}$ $(BC)^2 = (AC)^2+(AB)^2$

${(B C)}^{2} = {9.9}^{2} + {9.9}^{2}$ $(BC)^2=9.9^2 + 9.9^2$

Using the calculator,

${(B C)}^{2} = 98.01 + 98.01$ $(BC)^2=98.01+98.01$

${(B C)}^{2} = 196.02$ $(BC)^2=196.02$

$B C = \sqrt{196.02}$ $BC=sqrt(196.02$

$B C \approx 14.00071427$ $BC~~14.00071427$

$B C \approx 14.0$ $BC~~14.0$

Hence,

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

Hope it helps.

Answer 2 · 2018-08-06T02:23:34

14

Explanation:

The side is the square root of the area

$S \times S = A$ $S xx S = A$

S = $\sqrt{98}$ $sqrt 98$

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

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

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

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

this gives

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

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

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

$C = 14$ $C = 14$

The diagonal is 14

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

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