# Special lines in a triangle · Grade 10 · 4 units

> Special lines in a triangle, Grade 10 · 4 units: Where the medians meet and more. Explanation, worked examples and free practice following the Israeli curriculum.

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

- Example: (A triangle and its three medians, meeting at one point.) 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: 3 cm
- Solution: GM = 1/3 × 9 = 3 . It is 3 cm long.

### Medians and area

- Example: (A triangle and the median to its base. One half is shaded.) In triangle ABC , AD is the median to BC . The area of ABC is 24 cm². What is the area of triangle ABD ?
- Answer: 12 cm²
- Solution: Triangle ABD is half of ABC : 24 ÷ 2 = 12 . It is 12 cm².

### The median to the hypotenuse

- Example: (A right triangle with the median to the hypotenuse, dashed.) 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: 20 cm
- Solution: BM = 40 ÷ 2 = 20

### The angle bisector theorem

- Example: (A triangle, and a line from the top vertex to the base that bisects the angle.) 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: 5 cm
- Solution: By the angle bisector theorem, BD : DC = AB : AC = 12 : 15 . So DC = 4 × 15 ÷ 12 = 5 . It is 5 cm long.

### Loci

- Example: Which line holds every point that is the same distance from both ends of segment AB ?
- Options: The angle bisector · A median · An altitude · The perpendicular bisector
- Answer: The perpendicular bisector
- Solution: 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

- Example: A triangle has one point that is the same distance from all three vertices. Which point is it?
- Options: Where the angle bisectors meet · Where the medians meet · Where the altitudes meet · Where the perpendicular bisectors meet
- Answer: Where the perpendicular bisectors meet
- Solution: It is the point where the perpendicular bisectors meet.

## Not on the site 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. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 4 units topics](https://kixplo.com/il/en/math/grade-10-4u/)