Given the points A(−5 ; 5), B(1 ; −1) and C(3 ; 7).
1.1 Calculate the length of AB.
[ A 1.1 ]
1.2 Find the coordinates of D, the midpoint
of AB.
[ A 1.2 ]
1.3 Is CD perpendicular to AB? Give
a reason.
[ A 1.3 ]
1.4 Calculate the length of AC.
[ A 1.4 ]
1.5 Is ΔABC an isosceles triangle?
Give a reason.
[ A 1.5 ]
A(−3 ; 5), B(−7 ; −2) and C(1 ; −2) are the
vertices of ΔABC.
2.1 Calculate the folowing :
2.1.1 the gradient of BC.
[ A 2.1.1 ]
2.1.2 the length of AB.
[ A 2.1.2 ]
2.1.3 the coordinates of M, the
midpoint of BC.
[ A 2.1.3 ]
2.2 Is AM the perpendicular bisector
of BC? Give a reason.
[ A 2.2 ]
2.3 Is ∠ABC = ∠ACB? Give a reason.
[ A 2.3 ]
P(−6 ; 1), Q(2 ; −7) and R(8 ; −1) are the
vertices of ΔPQR.
3.1 Calculate teh coordinates of S, the
midpoint of PQ.
[ A 3.1 ]
3.2 Find the equation of the
median from R to PQ.
[ A 3.2 ]
3.3 Calculate the gradient of RQ.
[ A 3.3 ]
3.4 What kind of triangle is ΔPQR?
Give a reason.
[ A 3.4 ]
K(2 ; 9), L(−5 ; −5)
and M(−13 ; −1) are the
vertices of ΔKLM. Show that ΔKLM
is a right-angled triangle.
[ A 4.1 ]
D(5 ; 9), E(−3 ; 5) en F(−1 ; 1) are the
vertices of ΔDEF.
Show that ∠DEF = 90°.
[ A 5.1 ]
A(−6 ; 7), B(1 ; −3) and C(−9 ; −10) are the
vertices of ΔABC.
Show that ΔABC is an isosceles triangle.
[ A 6.1 ]
P(−3 ; −2), Q(−7 ; 7) and R(2 ; 3) are the
vertices of ΔPQR.
Show that ∠QPR = ∠PRQ.
[ A 7.1 ]
P(−4 ; 1), Q(4 ; −1) and R(5 ; −8) are the vertices
of ΔPQR and S is the point (1 ; −4).
8.1 Calculate the
8.1.1 coordinates of T, the midpoint
of PQ.
[ A 8.1.1 ]
8.1.2 equation of the median from R
from R to PQ.
[ A 8.1.2 ]
8.1.3 gradient of PR.
[ A 8.1.3 ]
8.2 Show that QS ⊥ PR.
[ A 8.2 ]
8.3 Calculate the area of ΔPQR
[ A 8.3 ]
A(−2 ; 3), B(−4 ; −1) and C(3 ; −2) are the
vertices of ΔABC and D is the
point (−3 ; 1).
9.1 Show that CD ⊥ AB.
[ A 9.1 ]
9.2 Find the equation of the
median from B to AC.
[ A 9.2 ]
9.3 Calculate the area of ΔABC
[ A 9.3 ]
The figure shows quadrilateral ABCD with
vertices A(−1 ; 4), B(−4 ; 1),
C(−1 ; −2) and D(3 ; 2).
10.1 Show that AB ∥ CD.
[ A 10.1 ]
10.2 Show that AB = CD.
[ A 10.2 ]
10.3 Calculate the size of ∠ABC.
[ A 10.3 ]
10.4 Stating reasons, say what kind of
quadrilateral ABCD is.
[ A 10.4 ]
The figure shows quadrilateral KLMN with
vertices K(1 ; 5), L(−3 ; −1), M(−1 ; −3)
and N(3 ; 3).
11.1 Show that KN ∥ LM.
[ A 11.1 ]
11.2 Show that KN = LM.
[ A 11.2 ]
11.3 Is KLMN a rectangle? Give a reason.
[ A 11.3 ]
The figure shows quadrialteral PQRS with
vertices P(2 ; 5), Q(−4 ; 1), R(−3 ; −2)
and S(3 ; 2) .
12.1 Calculate the coordinates of M, the
midpoint of PR.
[ A 12.1 ]
12.2 Show that QM = MS.
[ A 12.2 ]
12.3 Calculate the gradients of QR and
RS.
[ A 12.3 ]
12.4 Stating reasons, say what kind of
quadrilateral PQRS is.
[ A 12.4 ]
The figure shows quadrilateral ABCD with
vertics A(−2 ; 4), B(−6 ; 1), C(3 ; −1)
and D(7 ; 2).
13.1 Calculate the gradient of AD.
[ A 13.1 ]
13.2 Calculate the gradient of BC.
[ A 13.2 ]
13.3 Calculate the length of BC.
[ A 13.3 ]
13.4 Is AD = BC?
[ A 13.4 ]
13.5 Show that AC and BD bisect one
another at M.
[ A 13.5 ]
13.6 Give 2 reasons why ABCD
is a parallelogram.
[ A 13.6 ]
P(−2 ; 2), Q(1 ; −1), R(4 ; 2) and S(2 ; a) are the
vertices of quadrilateral PQRS. Calculate the
value of a if PS ∥ QR.
[ A 14. ]
K(−2 ; 3), L(2 ; 6), and M(5 ; 2) are the vertices
of ΔKLM.
15.1 Prove that ΔKLM is an isosceles,
right angled triangle,
[ A 15.1 ]
15.2 Calculate the area of ΔKLM.
[ A 15.2 ]
P(3 ; 2), Q(−3 ; 4), R(a ; b) en S(c ; d) are the
vertices of quadrilateral PQRS.
16.1 Calculate the coordinates of R if
T(1 ; 2) is the midpoint of PR.
[ A 16.1 ]
16.2 Calculate the coordinates of S if PQRS
is a parallelogram.
[ A 16.2 ]
16.3 Is PQRS a rectangle?
[ A 16.3 ]
A(−2 ; 1) and B(4 ; 16) are two points on
a line. C and D are two points on the line
with an equation of 5x − 2y − 14 = 0.
Can the quadrilateral ABCD be a
trapezium? Give reasons.
[ A 17. ]
P(−4 ; 13) and Q(2 ; −2) are two points
on a line. Q and R are two points on the line
line with an equation of 2x − 5y − 5 = 0.
Can PQ and QR be two adjacent sides of
a rectangle? Give reasons.
[ A 18. ]
A(2 ; 4), B(3 ; 1) C(6 ; 2) and D(5 ; 5);
are the vertices of quadrilateral ABCD
19.1 Calculate the length of AD.
[ A 19.1 ]
19.2 Calculate the coordinates of E the
midpoint of AC.
[ A 19.2 ]
19.3 Prove that DE = EB
[ A 19.3 ]
19.4 Say, without making any further
calculations and giving reasons,
which kind of quadrilateral can
quadrilateral ABCD be.
[ A 19.4 ]
19.5 Calculate the length of CD.
[ A 19.5 ]
19.6 Say, stating reasons, what kind of
quadrilateral can ABCD be.
[ A 19.6 ]
A(−3 ; 7), and B(4 ; −5) are two points.
From C(7 ; −1) a line CB is drawn so that.
∠ ABC = 90°.
20.1 Determine the equation of BC.
[ A 20.1 ]
21.2 Determine the length of AC.
[ A 20.2 ]
X(4 ; −3), Y(1 ; 2) and Z(6 ; 5) are the
vertices of ΔXYZ.
21.1 Determine the gradient of XY.
[ A 21.1 ]
21.2 Determine the lengte of XY.
[ A 21.2 ]
21.3 Determine the size of ∠YXZ.
[ A 21.3 ]
K(−6 ; 3), P(−3 ; 2) and M(9 ; −2) are three
points on the line with the equation
x + 3y - 3 = 0. M(−4 ; 3), P(−3 ; 2) and N(a ; b)
are points on another line.
22.1 Calculate the coordinates of N, i.e. the
values of a and b, if P is the midpoint
of LN.
[ A 22.1 ]
22.2 Calculate the lengths of KL and KN
[ A 22.2 ]
22.3 Show that KM is the perpendicular
bisector of LN.
[ A 22.3 ]
22.4 Stating reasons, say what kind of triangle
ΔKNP is.
[ A 22.4 ]
22.5 Calculate the length of LM.
[ A 22.5 ]
22.6 Stating reasons, say what kind of
quadrilateral KLMN is.
[ A 22.6 ]
A(−4 ; 2), and B(5 ; −4) are two points on a
straight line. C(−2 ; 4), and D(7 ; 6) are two
points on another straight line. Can AB
and CD be opposite sides of a parallelogram?
Give reasons for your answer.
[ A 23. ]
A(3 ; 4), C(5 ; 7) and B(p ; q) are three
points on a line. Calculate the coordinates
of B so that AC = CB
[ A 24. ]
K(−5 ; −4), and L(−1 ; 1) are two points on
a line. P(−3 ; −5), and Q(a ; b) are two points
on another line. Calculate the coordinates
of B so that
25.1 PQ ∥ KL.
[ A 25.1 ]
25.2 PQ ⊥ KL.
[ A 25.2 ]