Constructing a Tangent from an External Point
Aligned to G-C.A.4 — Common Core State Standards for Mathematics.
What this lesson teaches
From a point outside a circle, exactly two tangent lines touch the circle. The construction uses the fact that a tangent meets the radius at a right angle, so the tangent point lies on a circle with the segment from the center to the external point as diameter (Thales' theorem).
Worked example
Construct a tangent from external point P to a circle with center O. Draw segment OP and find its midpoint M. Draw a circle centered at M with radius MO; it passes through both O and P. This new circle crosses the original circle at point T. Because angle OTP is inscribed in a semicircle, it is 90°, so TP is perpendicular to radius OT — meaning TP is tangent.
Practice questions
- Why does the midpoint circle guarantee angle OTP is a right angle?
- How many tangent lines can be drawn to a circle from a point outside it?
- If OP = 10 and the radius OT = 6, use the right angle at T to find the tangent length PT.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Construir una tangente desde un punto externo
Desde un punto fuera de un círculo, exactamente dos líneas tangentes tocan el círculo. La construcción usa que una tangente forma un ángulo recto con el radio, así que el punto de tangencia está sobre un círculo cuyo diámetro es el segmento del centro al punto externo (teorema de Tales).
Ejemplo: Construye una tangente desde el punto externo P a un círculo con centro O. Traza el segmento OP y halla su punto medio M. Traza un círculo centrado en M con radio MO; pasa por O y por P. Este nuevo círculo corta el círculo original en el punto T. Como el ángulo OTP está inscrito en un semicírculo, mide 90°, así que TP es perpendicular al radio OT — es decir, TP es tangente.
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