The vertices of the square ABCD lie on a circle. point o lies on the side of AD and AO: OD = 1: 3.

| September 4, 2021 | education

The vertices of the square ABCD lie on a circle. point o lies on the side of AD and AO: OD = 1: 3. Ray CO intersects the circle at point P. Calculate the length of the chord CP if it is known that the area (S) of the square is 64 cm2.

Since ABCD is a square, then AB = BC = CD = AD = √Savsd = √64 = 8 cm.

Let’s draw a diagonal AC, then triangles АРС and АDC are rectangular, then triangle CDO is rectangular. Let us prove that triangles AOP and COD are similar.

Since the angle AOB = COD as vertical, the right-angled triangles AOP and COD are similar in acute angle.

Let us determine the lengths of the segments AO and DО. Since AO / OD = 1/3, then OD = 3 * OA.

ОD + ОА = АD = 8 cm, then 4 * ОА = 8 cm. ОА = 8/4 = 2 cm, then DO = 8 – 2 = 6 cm.

Then in a right-angled triangle COD, CO ^ 2 = OD ^ 2 + CD ^ 2 = 36 + 64 = 100.

CO = 10 cm.

From similar triangles AOP and COD

OS / AO = DO / RO.

10/2 = 6 / PO.

RO = 2 * 6/10 = 1.2 cm.

Then CP = OC + OP = 10 + 1.2 = 11.2 cm.

Answer: The length of the CP chord is 11.2 cm



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