# Find the value of x for each of the given figures?

Nov 10, 2016

a) x=6
b) x=12.8

#### Explanation:

a) x*15=5*(5+13)
$\implies x = \frac{5 \cdot 18}{15} = 6$

b) x*20=16^2
$\implies x = {16}^{2} / 20 = \frac{256}{20} = \frac{64}{5} = 12.8$

1) proof of the two-secant theorem:

Let $\angle S O Q = x$,
$P Q R S =$cyclic quadrilateral, => $\angle P S R = \angle P Q R = y$
$\implies \Delta O R Q \mathmr{and} \Delta O P S$ are similar,
$\implies \frac{O R}{O Q} = \frac{O P}{O S}$
$\implies \frac{c}{a + b} = \frac{a}{c + d}$
$\implies a \left(a + b\right) = c \left(c + d\right)$ proved.

2) proof of the tangent-secant theorem:

See Fig.1.
Let $T$ be the center of the circle.
$\angle O P T = {90}^{\circ}$
Let $\angle O P R = y , \implies \angle T P R = 90 - y = \angle T R P$
$\implies \angle P T R = 2 y , \implies \angle P S R = y$
$\implies \angle O P R = \angle P S R$, a well-known property of tangents

See Fig 2.
Let $\angle O P R = y , \implies \angle P S R = y$
Let $\angle P O S = x$
$\implies \Delta O P R \mathmr{and} \Delta O S P$ are similar.
$\implies \frac{O R}{O P} = \frac{O P}{O S}$
$\implies \frac{c}{a} = \frac{a}{c + d}$
${a}^{2} = c \left(c + d\right)$ proved.