Question

What are the coordinates of C ?

Answer 1

The figure shows a triangle ABC with one vertex B at the origin (0, 0). The base BC lies along the positive x-axis and has a length of 4 units. We need to find the coordinates of vertex C.

A point lying on the x-axis has a y-coordinate of 0. Its coordinates are of the form (x, 0).

Vertex B is at the origin (0, 0).
The side BC lies on the x-axis. This means the y-coordinate of point C must be 0.
The length of the segment BC is given as 4 units. Since B is at x=0, and C is on the positive x-axis, the x-coordinate of C will be 0 + 4 = 4.
Therefore, the coordinates of C are (4, 0).

The coordinates of C are (4, 0).

Answer 2

We need to find the altitude (height) of the equilateral triangle ABC from vertex A to the base BC.

In an equilateral triangle, the altitude bisects the base. This creates two congruent 30-60-90 right-angled triangles. We can use the Pythagorean theorem or the formula for the height of an equilateral triangle.
Formula for height (h): h = √(3)2 × side.
Pythagorean theorem: hypotenuse² = base² + height².

The triangle ABC is equilateral. The length of side BC is 4 units. Therefore, all sides (AB, BC, AC) are 4 units long.
Let the altitude from A meet BC at point D. In an equilateral triangle, this altitude is also the median, so D is the midpoint of BC.
The length of BD = (1)/(2) × BC = (1)/(2) × 4 = 2 units.
Now consider the right-angled triangle ADB.
Hypotenuse AB = 4.
Base BD = 2.
Height AD = ?
Using the Pythagorean theorem:
AB² = BD² + AD² 4² = 2² + AD² 16 = 4 + AD² AD² = 16 - 4 = 12 AD = √(12) = √(4 × 3) = 2√(3) Using the direct formula:
Side length s = 4.
Height h = √(3)2 × s = √(3)2 × 4 = 2√(3).

The height from A to BC is 2√(3) units.

Answer 3

We need to find the coordinates of the top vertex A of the equilateral triangle.

The coordinates of a point are given by its horizontal distance from the y-axis (x-coordinate) and its vertical distance from the x-axis (y-coordinate).
The x-coordinate of A will be the x-coordinate of the midpoint of the base BC.
The y-coordinate of A will be the height of the triangle, which we calculated in the previous part.

The base BC lies on the x-axis, with B at (0, 0) and C at (4, 0).
The altitude from A to BC bisects BC. The point where the altitude meets BC is the midpoint of BC.
Midpoint of BC = ( (0+4)/(2), (0+0)/(2) ) = ( (4)/(2), 0 ) = (2, 0).
The x-coordinate of vertex A must be the same as the x-coordinate of this midpoint. So, the x-coordinate of A is 2.
The y-coordinate of vertex A is its height above the x-axis. From part (ii), we calculated the height to be 2√(3).
Therefore, the coordinates of A are (2, 2√(3)).

The coordinates of A are (2, 2√(3)).