# QOTD-Geometry-Find the length of side SB

2019-07-03 | Team PendulumEdu

**A rectangular plane is cut into four triangles in such a way that the area of triangle APQ, ASB and BRQ is equal. Length of side BR is 4 cm. Find the length of SB.**

** A.** \((1+2\sqrt{5})m\)

** B.** \((\sqrt{5}-2)m\)

** C.** \(2(\sqrt{5}-1)m\)

** D.** \(2(1+\sqrt{5})m\)

**Solution**

Let the length of side SB = x

Length of side AS = y

Length of side PA = z

So, now \(PQ = x + 4\) and\( QR = y + z\)

According to the question

Area of triangle APQ = Area of triangle ASB = Area of triangle BRQ

\(\frac{1}{2}* z* (x+4)= \frac{1}{2} * y* x = \frac{1}{2}* (y+z)* 4\)

\(z*(x+4)= y+z* 4\)……………………………..(1)

\(z*x+4= y*x\)* *……………………………………….(2)

Using equation (1), we get

*\(4y+4z=zx+4z\)*

\(Z=\frac{4y}{x}\)

Now, substituting the value of z in equation (2), we get

\(\frac{4y}{x}*(x+4)=x*y\)

\(x^{2} - 4x - 16=0\)

\(x=2(1\pm\sqrt{5})\)

Length cannot be negative, therefore \( SB=2(1+\sqrt{5})\)

Hence, **(d)** is the correct option.

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