Understand tangent properties — a tangent is perpendicular to the radius, and tangents from an external point are equal in length.
A tangent to a circle at a point is always perpendicular to the radius drawn to that point.
A tangent touches a circle at point P. The radius OP is 5cm. If the tangent length from external point A is 12cm, find OA.
OA²=OP²+AP²=25+144=169; OA=13cm
Why must a tangent be perpendicular to the radius at the point of contact?
Because if it weren't, the tangent line would cross into the circle's interior somewhere, contradicting the definition of a tangent as touching at only one point
A tangent touches a circle at P with radius OP=9cm. If OA=15cm (A external), find the tangent length AP.
AP²=225-81=144; AP=12cm
Two tangents drawn from the same external point to a circle are always equal in length.
Two tangents from external point P touch a circle at A and B. If PA=8cm, find PB.
8cm, since tangents from the same external point are equal
A quadrilateral is circumscribed about a circle. What special property do its sides have?
The sums of opposite sides are equal (AB+CD=BC+DA), due to equal tangent segments from each vertex
A circle is inscribed in a quadrilateral with sides 6,8,10,8 (in order). Verify the opposite-side-sum property.
6+10=16, 8+8=16; equal, confirming the property holds for a tangential quadrilateral
1. A tangent touches a circle at P with radius OP=6cm. If OA=10cm (A external), find the tangent length AP.
2. Two tangents from external point Q touch a circle at C and D. If QC=11cm, find QD.
3. A circle is inscribed in a quadrilateral with sides 6,8,9,7. Verify the opposite-side-sum property (find if 6+9=8+7).
4. What angle does a tangent make with the radius at the point of contact?
5. From an external point 13cm from a circle's centre (radius 5cm), find the tangent length.
6. A tangent touches a circle at P with radius OP=8cm. If OA=17cm, find the tangent length AP.
7. Two tangents from external point Q touch a circle at C and D. If QC=14cm, find QD.
8. A circle is inscribed in a quadrilateral with sides 7,9,11,9 (in order). Verify the opposite-side-sum property.
9. What angle does a tangent make with the radius at the point of contact?
10. From an external point 25cm from a circle's centre (radius 7cm), find the tangent length.