Class 10 Tangents and Intersecting Chords Mock Test (ICSE)

Topics in Tangents and Intersecting Chords

• Tangent perpendicular to radius
• Tangents from an external point
• Alternate segment theorem
• Intersecting chords and secants
• Direct and transverse common tangents

Tangents and Intersecting Chords practice questions with answers

1. Q1Multiple choice·1 mark
   In a circle with centre $O$, $PA$ is a tangent to the circle at point $A$. If $\triangle OAP$ is a right-angled triangle with $\angle OPA = 35^\circ$, then $\angle AOP$ is:

   1. (a)$55^\circ$
   2. (b)$90^\circ$
   3. (c)$45^\circ$
   4. (d)$35^\circ$
   Show answer and step marks

   1. Correct option: (a) $55^\circ$+1

   Answer: (a)
2. Q2Multiple choice·1 mark
   A tangent $PQ$ touches a circle at $A$, and $AB$ is a chord. If $\angle BAT = 65^\circ$ (where $T$ is a point on $PQ$ on one side of $A$), the angle subtended by chord $AB$ in the alternate segment is:

   1. (a)$65^\circ$
   2. (b)$50^\circ$
   3. (c)$130^\circ$
   4. (d)$25^\circ$
   Show answer and step marks

   1. Correct option: (a) $65^\circ$+1

   Answer: (a)
3. Q3Multiple choice·1 mark
   Two circles of radii $10\text{ cm}$ and $2\text{ cm}$ have their centres $17\text{ cm}$ apart. The length of their direct common tangent is:

   1. (a)$15\text{ cm}$
   2. (b)$13\text{ cm}$
   3. (c)$19\text{ cm}$
   4. (d)$17\text{ cm}$
   Show answer and step marks

   1. Correct option: (a) $15\text{ cm}$+1

   Answer: (a)
4. Q4Multiple choice·1 mark
   From an external point $P$, two tangents $PA$ and $PB$ are drawn to a circle with centre $O$. If $\angle APB = 70^\circ$, then $\angle OAB$ is:

   1. (a)$35^\circ$
   2. (b)$55^\circ$
   3. (c)$45^\circ$
   4. (d)$70^\circ$
   Show answer and step marks

   1. Correct option: (a) $35^\circ$+1

   Answer: (a)
5. Q5Long answer·4 marks
   Two circles with centres $A$ and $B$ have radii $8\text{ cm}$ and $2\text{ cm}$ respectively. The distance between their centres $AB$ is $2\sqrt{34}\text{ cm}$. (a) Calculate the length of their direct common tangent. (b) Calculate the length of their transverse common tangent.
   Show answer and step marks

   1. (a) Formula for the length of the direct common tangent: $\sqrt{d^2 - (R - r)^2}$, with $R = 8\text{ cm}$, $r = 2\text{ cm}$, $d = 2\sqrt{34}\text{ cm}$.+1
   2. Substitute: $\sqrt{(2\sqrt{34})^2 - (8 - 2)^2} = \sqrt{136 - 36} = \sqrt{100} = 10\text{ cm}$.+1
   3. (b) Formula for the length of the transverse common tangent: $\sqrt{d^2 - (R + r)^2}$.+1
   4. Substitute: $\sqrt{(2\sqrt{34})^2 - (8 + 2)^2} = \sqrt{136 - 100} = \sqrt{36} = 6\text{ cm}$.+1

   Answer: (a) $10\text{ cm}$, (b) $6\text{ cm}$

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