Question

In Figure, lines XY and MN intersect at O. If \(∠\)POY=90° and a:b=2:3, find c.

Answer 1

Given: 

\[ \angle POY = 90^\circ \quad \text{and} \quad \frac{a}{b} = \frac{2}{3} \]

Let the common ratio between \(a\) and \(b\) be \(x\).

Thus, XY is a straight line, and rays OM and OP stand on it.

We have:

\[ \angle POY = \angle POX = 90^\circ \]

From the angle equation:

\[ \angle POX = \angle POM + \angle MOX \]

So,

\[ 90^\circ = a + b \]

Now, \(a = 2x\) and \(b = 3x\), therefore:

\[ a + b = 90^\circ \]

Substituting \(a = 2x\) and \(b = 3x\):

\[ 2x + 3x = 90^\circ \]

\[ 5x = 90^\circ \]

\[ x = \frac{90^\circ}{5} = 18^\circ \]

Thus,

\[ a = 2x = 2 \times 18^\circ = 36^\circ \]

\[ b = 3x = 3 \times 18^\circ = 54^\circ \]

Also,

\[ \angle MOY = \angle MOP + \angle POY = a + 90^\circ \]

\[ \angle MOY = 36^\circ + 90^\circ = 126^\circ \]

Therefore, \(c = 126^\circ\).