Special lines in a triangle · Grade 10 · 4 units
In Special lines in a triangle (Grade 10 · 4 units) you practise 6 skills: Where the medians meet · Medians and area · The median to the hypotenuse · The angle bisector theorem · Loci · The special points of a triangle. Each special line in a triangle is a locus. The perpendicular bisector of a segment is the set of points equally far from its ends, and an angle bisector is the set of points equally far from the arms of the angle. The three perpendicular bisectors meet at one point, the centre of the circumscribed circle, and the three angle bisectors meet at the centre of the inscribed circle.
The three medians meet at one point, which divides each median in the ratio two to one from the vertex. A median splits the triangle into two triangles of equal area, because they have equal bases and the same height. In a right triangle the median to the hypotenuse is half the hypotenuse. An angle bisector divides the opposite side in the ratio of the two sides next to the angle. A model proof: take a point on the perpendicular bisector of a segment. With the ends of the segment it makes two triangles that share a side, the bisector, and a right angle. The parts next to the right angle are equal too, because the bisector passes through the midpoint. So the triangles are congruent by SAS, and the point is equally far from both ends.
Tip: Draw the special line in another colour, and mark what you know about it: a midpoint, a right angle or two equal angles.
What you learn
Where the medians meet
For example: In triangle ABC , the medians meet at G . The line AM is a median. The median AM is 9 cm long. How long is GM ?
Answer and solution
The answer: 3 cm
GM = 1/3 × 9 = 3 . It is 3 cm long.
Medians and area
For example: In triangle ABC , AD is the median to BC . The area of ABC is 24 cm². What is the area of triangle ABD ?
Answer and solution
The answer: 12 cm²
Triangle ABD is half of ABC : 24 ÷ 2 = 12 . It is 12 cm².
The median to the hypotenuse
For example: In the triangle ABC has ∢B = 90° , and BM is the median to the hypotenuse AC . The hypotenuse is 40 cm. How long is the median?
Answer and solution
The answer: 20 cm
BM = 40 ÷ 2 = 20
The angle bisector theorem
For example: In triangle ABC , AD bisects ∢BAC . Point D lies on side BC . In centimetres, AB = 12, AC = 15, BD = 4 . How long is DC ?
Answer and solution
The answer: 5 cm
By the angle bisector theorem, BD : DC = AB : AC = 12 : 15 . So DC = 4 × 15 ÷ 12 = 5 . It is 5 cm long.
Loci
For example: Which line holds every point that is the same distance from both ends of segment AB ?
- The angle bisector
- A median
- An altitude
- The perpendicular bisector
Answer and solution
The answer: The perpendicular bisector
The perpendicular bisector of a segment is the locus of all the points the same distance from both its ends.
The special points of a triangle
For example: A triangle has one point that is the same distance from all three vertices. Which point is it?
- Where the angle bisectors meet
- Where the medians meet
- Where the altitudes meet
- Where the perpendicular bisectors meet
Answer and solution
The answer: Where the perpendicular bisectors meet
It is the point where the perpendicular bisectors meet.
Not on the site yet
Parts of this topic that are not built yet: Proving that the special lines meet · Constructions with a ruler and compasses · Proofs with special lines.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).