Angle Bisectors and Perpendicular Bisectors Worksheet
A free, printable geometry practice worksheet on Angle Bisectors and Perpendicular Bisectors with the full answer key included — download it, print it, assign it. Pair it with the matching video lesson below.
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Every Numberbender worksheet is paired with a full video lesson by Dr. Peter Esperanza — watch the concept explained, then work through the sheet and check yourself with the answer key.
Angle bisectors vs. perpendicular bisectors
Both are “bisectors” — they cut something exactly in half — but they bisect different things:
- Angle bisector: a ray that divides an angle into two equal angles. Every point on it is the same distance from the two sides of the angle.
- Perpendicular bisector: a line that crosses a segment at its midpoint at a right angle. Every point on it is the same distance from the segment’s two endpoints.
The Angle Bisector Theorem
In a triangle, an angle bisector splits the opposite side into pieces proportional to the two adjacent sides. If \(AD\) bisects \(\angle A\) in \(\triangle ABC\) and meets \(BC\) at \(D\), then:
$$\dfrac{BD}{DC}=\dfrac{AB}{AC}$$
The Perpendicular Bisector Theorem
A point lies on the perpendicular bisector of \(\overline{AB}\) if and only if it is equidistant from \(A\) and \(B\), that is \(PA = PB\). This is why the three perpendicular bisectors of a triangle’s sides meet at the circumcenter — the one point equally far from all three vertices.
Worked example
Problem. In \(\triangle ABC\), the bisector of \(\angle A\) meets \(BC\) at \(D\). Given \(AB = 8\), \(AC = 6\), and \(BC = 7\), find \(BD\) and \(DC\).
Solution. By the Angle Bisector Theorem, \(\dfrac{BD}{DC}=\dfrac{AB}{AC}=\dfrac{8}{6}=\dfrac{4}{3}\). Split \(BC = 7\) in the ratio \(4:3\): \(BD = \dfrac{4}{7}\times 7 = 4\) and \(DC = \dfrac{3}{7}\times 7 = 3\).
Answer: \(BD = 4\), \(DC = 3\). See the full step-by-step →
Frequently asked questions
What’s the difference between an angle bisector and a perpendicular bisector?
An angle bisector divides an angle into two equal angles; a perpendicular bisector divides a segment in half at a 90° angle. One works on angles, the other on segments.
What is the Angle Bisector Theorem?
In a triangle, the bisector of an angle divides the opposite side into two segments proportional to the other two sides: \(BD/DC = AB/AC\).
What is the Perpendicular Bisector Theorem?
Any point on the perpendicular bisector of a segment is equidistant from the segment’s two endpoints — and the reverse is true too.
Where do the perpendicular bisectors of a triangle meet?
They meet at the circumcenter, the single point equidistant from all three vertices, which is the center of the triangle’s circumscribed circle.
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