Using the origin as one vertex, plot the vertices of:
- (i) A right-angled isosceles triangle.
- (ii) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.
Step-by-step solution
Idea: The axes meet at a right angle, so two equal lengths marked along the two axes from O give a right-angled isosceles triangle. Points that are mirror images in the y-axis are equally far from O, which gives an isosceles triangle across Quadrants III and IV.
(i) A right-angled isosceles triangle.
- Take A (4, 0) on the x-axis and B (0, 4) on the y-axis. The axes are perpendicular, so ∠AOB = 90°.½ mark
- OA = 4 and OB = 4, so the triangle is isosceles; its hypotenuse is AB = √(42 + 42) = 4√2. Vertices O (0, 0), A (4, 0), B (0, 4).½ mark
(ii) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.
- Take P (−3, −4) in Quadrant III and its mirror image in the y-axis, Q (3, −4), in Quadrant IV.½ mark
- OP = √((−3)2 + (−4)2) = 5 and OQ = √(32 + (−4)2) = 5, so OP = OQ and triangle OPQ is isosceles (PQ = 6). Vertices O (0, 0), P (−3, −4), Q (3, −4).½ mark
Check: (i) AB2 = 32 = OA2 + OB2 = 16 + 16 ✓, so the right angle is at O. (ii) P is (−, −), Quadrant III ✓; Q is (+, −), Quadrant IV ✓.
Answer to write in the exam
(i)
O (0, 0), A (4, 0), B (0, 4)
OA = 4, OB = 4; ∠AOB = 90° (axes are perpendicular)
AB = √(42 + 42) = 4√2
∴ OAB is a right-angled isosceles triangle.
(ii)
O (0, 0), P (−3, −4) (Quadrant III), Q (3, −4) (Quadrant IV)
OP = √(9 + 16) = 5, OQ = √(9 + 16) = 5
∴ OP = OQ, so OPQ is isosceles.
Common mistakes that cost marks
- In (i), taking A (4, 0) and B (0, 3). The legs must be equal for the triangle to be isosceles.
- In (ii), choosing P (−3, −4) and Q (2, −4) without checking: OP = 5 but OQ = √20, so the triangle is not isosceles. Mirror images in the y-axis are always equally far from O.
- Putting the second vertex on an axis, e.g. (0, −4). A point on an axis is not in any quadrant.
How this can come in the exam
The points O (0, 0), A (6, 0) and B (0, 6) form
- a right-angled isosceles triangle
- an equilateral triangle
- a scalene triangle
- an obtuse-angled triangle
Show answer
(A) a right-angled isosceles triangle
OA = OB = 6 and the axes are perpendicular, so the angle at O is 90°.
Show that O (0, 0), P (−2, −7) and Q (2, −7) form an isosceles triangle, and find its perimeter.
Show answer
OP = √(4 + 49) = √53 and OQ = √(4 + 49) = √53, so OP = OQ (1 mark). PQ = 2 − (−2) = 4. Perimeter = 4 + 2√53 units (1 mark).Try one yourself
Using the origin as one vertex, give an isosceles triangle with its other two vertices in Quadrant I and Quadrant II.
Show answer
For example O (0, 0), (4, 3), (−4, 3): both slanting sides are √(16 + 9) = 5.
More questions like this
- The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer.
- Use the connection you found to find the coordinates of B given that M (−7, 1) is the midpoint of A (3, −4) and B (x, y).
- Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find the coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, −2).
- (i) Given the points A (1, −8), B (−4, 7) and C (−7, −4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
(ii) Given the points D (−5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K. - The midpoints of the sides of triangle ABC are the points D, E, and F. Given that the coordinates of D, E, and F are (5, 1), (6, 5), and (0, 3), respectively, find the coordinates of A, B and C.