Circles: math test, Grade 10 – download the PDF

1 Tangents / 3 pts

A circle has center \(O\) and radius \(9\) cm. The point \(P\) is \(15\) cm from \(O\). Two tangent segments \(PT\) and \(PS\) are drawn to the circle. (a) Find \(PT\). (b) What is \(PS\)? Explain.

2 Inscribed angles / 4 pts

Points \(A, B, C, D\) lie on a circle and split it into arcs \(\widehat{AB} = 84^\circ\), \(\widehat{BC} = 96^\circ\), \(\widehat{CD} = 70^\circ\), \(\widehat{DA} = 110^\circ\) (see the figure).

(a) Find \(m\angle ACB\). (b) Find \(m\angle ABD\). (c) Find \(m\angle ABC\) and explain what it tells you about \(AC\). (d) Find \(m\angle DAB\) and \(m\angle BCD\), and show that they are supplementary.

3 Arc and sector / 4 pts

A circle has radius \(15\) cm. A sector has a central angle of \(48^\circ\). Find (a) the length of the arc and (b) the area of the sector. Give exact values, then round to the nearest hundredth. Calculator allowed.

4 Segment lengths / 4 pts

(a) Chords \(AB\) and \(CD\) meet at \(E\) with \(AE = 8\), \(EB = 3\), \(CE = 4\). Find \(ED\). (b) From a point \(P\), a tangent of length \(6\) and a secant with external part \(4\) are drawn. Find the length of the whole secant and of the chord inside the circle.

5 Equation of a circle / 3 pts

Consider \(x^2 + y^2 - 12x + 4y + 15 = 0\). (a) Rewrite it in standard form. (b) Give the center and radius. (c) Does the point \((2, 1)\) lie on the circle?

6 Angle outside a circle / 2 pts

Two secants meet at a point outside a circle. The far arc measures \(130^\circ\) and the near arc measures \(46^\circ\). Find the measure of the angle formed by the secants.