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CBSE Class 9 Triangles, Circles and Geometry — practice questions

Explore 50 free Triangles, Circles and Geometry questions in CBSE Class 9 Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Triangles, Circles and Geometry questions and exercises with answers

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If two angles of one triangle are respectively equal to two angles of another triangle, then the triangles are similar by theRead the answer and worked solution →If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at an angle of $80^\circ$, then $\angle POA$ isRead the answer and worked solution →From a point $Q$, the length of the tangent to a circle is $24\text{ cm}$ and the distance of $Q$ from the centre is $25\text{ cm}$. The radius of the circle isRead the answer and worked solution →The common point of a tangent and a circle is called theRead the answer and worked solution →If XY is a tangent to a circle with centre O at point P, then $OP$ isRead the answer and worked solution →At a given point on a circle, how many tangents can be drawn?Read the answer and worked solution →A line that intersects a circle at exactly two distinct points is called aRead the answer and worked solution →As per your syllabus, given a triangle with sides a, b, c — what is the SEMI-PERIMETER s used in Heron's formula?Read the answer and worked solution →As per your syllabus, what is HERON's FORMULA for the area of a triangle with sides a, b, c and semi-perimeter s?Read the answer and worked solution →As per your syllabus, 's Example 5, applying Heron's formula to a triangle with sides 3, 4, 5 gives what area?Read the answer and worked solution →As per your syllabus, for a triangle with sides a, b, c and CIRCUMCIRCLE radius R, what is the area formula?Read the answer and worked solution →As per your syllabus, for a triangle with sides a, b, c and INCIRCLE radius r, the area equals which expression?Read the answer and worked solution →As per your syllabus, what is BRAHMAGUPTA's FORMULA for the area of a CYCLIC quadrilateral with sides a, b, c, d and semi-perimeter s?Read the answer and worked solution →As per your syllabus, a CYCLIC quadrilateral has sides 3, 4, 5, 6. What is its area (using Brahmagupta's formula)?Read the answer and worked solution →Two triangles are similar if:Read the answer and worked solution →By the Pythagoras theorem, in a right-angled triangle:Read the answer and worked solution →A tangent to a circle is:Read the answer and worked solution →From a point outside a circle, the two tangents drawn are:Read the answer and worked solution →CLASS9 · Mathematics · Triangle Property Gap FillRead the answer and worked solution →CLASS10 · Mathematics · Similarity Criteria — TrianglesRead the answer and worked solution →CLASS10 · Mathematics · Circles — True or FalseRead the answer and worked solution →CLASS10 · Mathematics · Tap the PolygonsRead the answer and worked solution →SAT · Mathematics · Triangle Similarity — T/FRead the answer and worked solution →SAT · Mathematics · Circles — Arc & SectorRead the answer and worked solution →Match the polygon to its sum of interior anglesRead the answer and worked solution →Sort: triangle inequality satisfied or not?Read the answer and worked solution →Match the quadrilateral to its defining featureRead the answer and worked solution →If the angles of a triangle are in the ratio 1 : 2 : 3, then what is the smallest angle?Read the answer and worked solution →The ratio of the interior to exterior angle of a regular polygon is 4:1. How many sides does the polygon have?Read the answer and worked solution →A right triangle has legs 6 and 8 cm. Hypotenuse:Read the answer and worked solution →

Worked practice questions

If two angles of one triangle are respectively equal to two angles of another triangle, then the triangles are similar by the

  1. SSS similarity criterion
  2. SAS similarity criterion
  3. AA similarity criterion
  4. RHS congruency criterion
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Correct answer: C. AA similarity criterion

If two corresponding angles are equal, then the third angles are also equal by the angle sum property of a triangle. **Hence the triangles are similar by the AA criterion.**

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If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at an angle of $80^\circ$, then $\angle POA$ is

  1. $50^\circ$
  2. $60^\circ$
  3. $70^\circ$
  4. $80^\circ$
Show answer and solution

Correct answer: A. $50^\circ$

Since tangents from an external point are equal, $PA = PB$, and also $OA = OB$. Hence triangles $OAP$ and $OBP$ are congruent, so $OP$ bisects $\angle APB$. Therefore $\angle APO = 40^\circ$. In right triangle $OAP$, $\angle OAP = 90^\circ$ because radius is perpendicular to tangent so $\angle POA = 180^\circ - 90^\circ - 40^\circ = 50^\circ$.

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From a point $Q$, the length of the tangent to a circle is $24\text{ cm}$ and the distance of $Q$ from the centre is $25\text{ cm}$. The radius of the circle is

  1. $7\text{ cm}$
  2. $12\text{ cm}$
  3. $15\text{ cm}$
  4. $24.5\text{ cm}$
Show answer and solution

Correct answer: A. $7\text{ cm}$

The radius to the point of contact is perpendicular to the tangent, so the triangle formed is right-angled. If radius is $r$, then $25^2 = 24^2 + r^2$. Thus $r^2 = 625 - 576 = 49$, so $r = 7\text{ cm}$. Options B, C and D do not satisfy the Pythagorean relation.

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