Question

Can someone help me on this geometry problem & explain ?

Answer 1

Hey there!!

We know :

... The given shape being a square, we know all the sides have the equal measurements and each of the angle is 90 degrees. Hence, a square could be broken down into two right triangles. The sides would be counted as the legs and the diagonal is written as the hypotenuse of the right triangle. We can derive the measures using Pythagorean  theorem. In a square the diagonals bisect each other.

What is the pythagorean theorem ?

... a² + b² = c²

( Where the a and the b are the legs and the c is the hypotenuse )

Let's take the measure of the side as 'x'

We know the diagonal or the hypotenuse is 14

... x² + x² = 14²

... 2x² = 196

... x² = 98  

... x = √98

... x = 7√2

Hence, the measure of each side is 7√2.

QR = 7√2

As we know the diagonals bisect each other, the measure of ST would be 14 / 2

Hence, ST = 7

Hope my answer helps!!

Answer 2

If PQRS is a square.

Then |SP| = |PQ| = |QR| = |SR| and |PR| = |SQ|.

$\triangle PSR\ \text{is a right triangle}.\\\\\text{Let}\ |SP|=|SR|=x.\\\\\text{We can use the Pythagorean theorem:}\\\\|SP|^2+|SR|^2=|PR|^2\\\\\text{Substitute}\\\\x^2+x^2=14^2\\\\2x^2=196\qquad|\text{divide both sides by 2}\\\\x^2=98\to x=\sqrt{98}\\\\x=\sqrt{49\cdot2}\\\\x=\sqrt{49}\cdot\sqrt2\\\\\boxed{x=7\sqrt2}$

$|ST|=\dfrac{1}{2}|PR|\to|ST|=\dfrac{1}{2}\cdot14=7$

$Answer:\ ST=7;\ QR=7\sqrt2$

You can use the formula of a diagonal of a square

a - side of a square

d - diagonal of a square

$d=a\sqrt2$

d = 14

$a\sqrt2=14\qquad|\cdot\sqrt2\\\\2a=14\sqrt2\qquad|:2\\\\a=7\sqrt2$