Embedded Files
STEP 2001, Math I, #1: The points A, B, and C lie on the sides of a square of side 1 cm and no two points lie on the same side.
Show that the length of at least one side of the triangle ABC must be less than or equal to
cm.
Red Question: (PHTST) A rectangle with vertices A(-6, 1) , B(-2, 1) , C(-2 , 7) , and D(-6 , 7) is rotated
clockwise about the
point (- 6 , 1). What is the area of the portion of this rotated rectangle that lies above the x-axis?
STEP 2009, Math I, #4: The sides of a triangle have lengths
, and , where . The largest and smallest
angles of the triangle are
and , respectively. Show by means of the cosine rule that
In the case
, show that and hence find the ratio of the lengths of the sides of the triangle.
A complete solution to this problem is posted below.
STEP 2005, Math II, #5: The angle A of triangle ABC is a right angle and the sides BC, CA and AB are of lengths
, and ,
respectively. Each side of the triangle is tangent to the circle
which is of radius . Show that . (Hint: There are
several ways to establish this first result. You may find it easiest to first establish the fact that
.)
Each vertex of the triangle lies on the circle . The ratio of the area of the region between and the triangle to the area of is
denoted by
. Show that
,
where
.
Deduce that
STEP 2006, Math I, #2: A small goat is tethered by a rope to a point at ground level on a side of a square barn which stands in a
large horizontal field of grass. The sides of the barn are of length
and the rope is of length . Let be the area of the grass
that the goat can graze.
Prove that
and determine the minimum value of
.
STEP 2000, Math II, #3: The lengths of the sides BC, CA, AB of the triangle ABC are denoted by
, , and , respectively.
Given that
, , angle
where
, and are small, show that , where .
[Hint: Perhaps useful to your argument can be the following approximations:
When
is very small, close to 0, then
and
Also,
]
Given now that
, , ,
find the range of possible values of
.
STEP 2012, Math II, #6: A cyclic quadrilateral ABCD has sides AB, BC, CD and DA of lengths
, , and , respectively.
The area of the quadrilateral is
, and angle DAB is . [Hint: Think about what is special about a cyclic quadrilateral. Draw one
inscribed within a circle.]
Find an expression for
in terms of , , and , and an expression for in terms of , , , and . Hence show that
,
and deduce that
where
Deduce a formula for the area of a triangle with sides of length
, and .
STEP 2004, Math I, #6: The three points A, B, and C have coordinates , , and , respectively. Find the point of
intersection of the line joining A to the midpoint of BC, and the line joining B to the midpoint of AC. Verify that this point lies on the line
joining C to the midpoint of AB.
The point H has coordinates
. Show that if the line AH intersects the line BC at right angles, then ,
and write down a similar result if the line BH intersects the line AC at right angles.
Deduce that if AH is perpendicular to BC and also BH is perpendicular to AC, then CH is perpendicular to AB.
STEP 2009, Math II, #1: Two curves have equations and , where and are positive constants. State the equation
of the lines of symmetry of each curve.
The curves intersect at the distinct points A, B, C, D (taken anticlockwise from A). The coordinates of A are , where .
Write down, in terms of
and , the coordinates of B, C and D.
Show that the quadrilateral ABCD is a rectangle and find its area in terms of
and only. Verify that, for the case and ,
the area is 14.
STEP 1999, Math I, #2: A point moves in the xy-plane so that the sum of the squares of its distances from the three fixed points ,
, and is always . Find the equation of the locus of the point and interpret it geometrically. Explain why cannot be
less than the sum of the squares of the distances of the three points from their centroid.
[The centroid has coordinates where , .]
STEP 2009, Math I, #8: (i) The equation of the circle C is
,
where
is a positive number. Show that C touches the line .
Let
be the acute angle between the x-axis and the line joining the origin to the centre of C. Show that
and deduce that C touches the line .
(ii) Find the equation of the incircle of the triangle formed by the lines , and .
[Note: The incircle of a triangle is the circle, lying totally inside the triangle, that touches all three sides.]
STEP 2009, Math III, #1: The points S, T, U and V have coordinates , , and , respectively. The lines
SV and UT meet the line
at the points with coordinates and , respectively. Show that
and write down a similar expression for
.
Given that S and T lie on the circle , find a quadratic equation satisfied by and , and hence determine and
in terms of
, and .
Given that S, T, U and V lie on the above circle, show that .
STEP 2004, Math II, #4:
(i) An attempt is made to move a rod of length L from a corridor of width
into a corridor of width , where . The corridors meet
at right angles, as sown in Figure 1 and the rod remains horizontal. Show that if the attempt is to be successful then
where
satisfies
(ii) An attempt is made to move a rectangular table-top, of width
and length , from one corridor to the other, as shown in Figure 2.
The table-top remains horizontal. Show that if the attempt is to be successful then
where satisfies
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