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PSLE

PSLE Foundation Mathematics

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<p>The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.</p>
<ol class="abc-list"><li data-list-item-id="cg-q13-a" data-list-item-path="cg-q13-a"><div class="list-item-paragraph">Find the equation of the perpendicular bisector of $AB$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q13-b" data-list-item-path="cg-q13-b"><div class="list-item-paragraph">Find the coordinates of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q13-c" data-list-item-path="cg-q13-c"><div class="list-item-paragraph">Find the area of the triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.

Find the equation of the perpendicular bisector of $AB$.

Area of Rectilinear Figures

Parallel and Perpendicular Lines

Intersection of Two Lines

Equation of a Straight Line

Midpoint of a Line Segment

Perpendicular Bisectors

Perpendicular bisector and the third vertex of a triangle

Coordinate Geometry

<p>The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$, $(k,\ 0)$ and $(15,\ 4)$ respectively. $AB$ cuts the $y$-axis at $P$. The point $Q$ lies on $AC$ and $PQ$ is parallel to $BC$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q10-i" data-list-item-path="cg-q10-i"><div class="list-item-paragraph">Given that&nbsp;$k&lt;5$, find the value of $k$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-ii" data-list-item-path="cg-q10-ii"><div class="list-item-paragraph">Find the coordinates of $P$ and show that&nbsp;$P$ is the midpoint of $AB$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-iii" data-list-item-path="cg-q10-iii"><div class="list-item-paragraph">Hence find the coordinates of $Q$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q10-iv" data-list-item-path="cg-q10-iv"><div class="list-item-paragraph">The point $D$ is such that&nbsp;$ABCD$ is a rectangle. Find the coordinates of $D$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q10-v" data-list-item-path="cg-q10-v"><div class="list-item-paragraph">Find the area of the rectangle $ABCD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q10-vi" data-list-item-path="cg-q10-vi"><div class="list-item-paragraph">The point $F$ lies on $BC$ such that&nbsp;$BF : FC = 1 : 3$. Find the coordinates of $F$.<span class="inline-marks">[2]</span></div></li></ol>

The figure shows a right-angled triangle $ABC$ in which $A$, $B$ and $C$ are $(-3,\ 18)$, $(k,\ 0)$ and $(15,\ 4)$ respectively. $AB$ cuts the $y$-axis at $P$. The point $Q$ lies on $AC$ and $PQ$ is parallel to $BC$.

Given that $k<5$, find the value of $k$.

Find the coordinates of $P$ and show that $P$ is the midpoint of $AB$.

The point $D$ is such that $ABCD$ is a rectangle. Find the coordinates of $D$.

The point $F$ lies on $BC$ such that $BF : FC = 1 : 3$. Find the coordinates of $F$.

Ratio Theorem and Section Formula

Right-angled triangle completed to a rectangle

<p>$ABCD$ is a rhombus in which $A$ is $(4,\ 6)$ and $B$ is $(5,\ 14)$. The diagonals $AC$ and $BD$ intersect at $T$, and $AC$ cuts the $y$-axis at $(0,\ 4)$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q07-i" data-list-item-path="cg-q07-i"><div class="list-item-paragraph">Find the equations of $AC$ and $BD$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-ii" data-list-item-path="cg-q07-ii"><div class="list-item-paragraph">Calculate the coordinates of $T$, $C$ and $D$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-iii" data-list-item-path="cg-q07-iii"><div class="list-item-paragraph">Find the area of the rhombus $ABCD$.<span class="inline-marks">[3]</span></div></li></ol>

$ABCD$ is a rhombus in which $A$ is $(4,\ 6)$ and $B$ is $(5,\ 14)$. The diagonals $AC$ and $BD$ intersect at $T$, and $AC$ cuts the $y$-axis at $(0,\ 4)$.

Calculate the coordinates of $T$, $C$ and $D$.

Diagonals of a rhombus

<p>The diagram shows a kite $ABCD$ in which the side $AB$ is parallel to the $x$-axis. The diagonals $AC$ and $BD$ intersect at $T$.</p>
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<ol class="i-list"><li data-list-item-id="cg-q06-i" data-list-item-path="cg-q06-i"><div class="list-item-paragraph">It is given that&nbsp;$B$ is $(7,\ 2)$ and the equation of the diagonal $AC$ is $2y=x+7$.</div><ol class="abc-list"><li data-list-item-id="cg-q06-i-a" data-list-item-path="cg-q06-i/cg-q06-i-a"><div class="list-item-paragraph">Find the coordinates of $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q06-i-b" data-list-item-path="cg-q06-i/cg-q06-i-b"><div class="list-item-paragraph">Find the equation of the diagonal $BD$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q06-i-c" data-list-item-path="cg-q06-i/cg-q06-i-c"><div class="list-item-paragraph">Hence find the coordinates of $T$ and of $D$.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="cg-q06-ii" data-list-item-path="cg-q06-ii"><div class="list-item-paragraph">Given also that the coordinates of $C$ are $(11,\ 9)$, find the area of the kite $ABCD$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a kite $ABCD$ in which the side $AB$ is parallel to the $x$-axis. The diagonals $AC$ and $BD$ intersect at $T$.

It is given that $B$ is $(7,\ 2)$ and the equation of the diagonal $AC$ is $2y=x+7$.

Given also that the coordinates of $C$ are $(11,\ 9)$, find the area of the kite $ABCD$.

Kite with a side parallel to the x-axis

<p>The diagram shows a triangle $ABC$ in which $A$ is $(0,\ 6)$, $B$ is $(a,\ 7)$ and $C$ is $(12,\ 0)$. The point $M$ is the midpoint of $AC$ and $BM$ is perpendicular to $AC$.</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q05-i" data-list-item-path="cg-q05-i"><div class="list-item-paragraph">the coordinates of $M$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q05-ii" data-list-item-path="cg-q05-ii"><div class="list-item-paragraph">the equation of the perpendicular bisector of $AC$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q05-iii" data-list-item-path="cg-q05-iii"><div class="list-item-paragraph">the value of $a$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q05-iv" data-list-item-path="cg-q05-iv"><div class="list-item-paragraph">the area of the triangle $ABC$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a triangle $ABC$ in which $A$ is $(0,\ 6)$, $B$ is $(a,\ 7)$ and $C$ is $(12,\ 0)$. The point $M$ is the midpoint of $AC$ and $BM$ is perpendicular to $AC$.

the equation of the perpendicular bisector of $AC$,

Perpendicular bisector of a side of a triangle

<p>The diagram shows a rectangle $ABCD$ in which $A$ and $C$ lie on the $x$-axis. The equation of $AB$ is $3y=x+1$ and the equation of $BD$ is $4y=3x-12$.</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q04-i" data-list-item-path="cg-q04-i"><div class="list-item-paragraph">the coordinates of $B$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-ii" data-list-item-path="cg-q04-ii"><div class="list-item-paragraph">the coordinates of the midpoint of $AC$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-iii" data-list-item-path="cg-q04-iii"><div class="list-item-paragraph">the coordinates of $A$, $C$ and $D$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q04-iv" data-list-item-path="cg-q04-iv"><div class="list-item-paragraph">the length of $BC$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-v" data-list-item-path="cg-q04-v"><div class="list-item-paragraph">the area of the rectangle $ABCD$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a rectangle $ABCD$ in which $A$ and $C$ lie on the $x$-axis. The equation of $AB$ is $3y=x+1$ and the equation of $BD$ is $4y=3x-12$.

the coordinates of the midpoint of $AC$,

the coordinates of $A$, $C$ and $D$,

Length of a Line Segment

Rectangle with a diagonal along the x-axis

<p>The diagram shows a rectangle $ABCD$ in which $A$ is $(1,\ 2)$ and $D$ is $(5,\ 0)$. The diagonal $AC$ lies on the line $y=x+1$.</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q03-i" data-list-item-path="cg-q03-i"><div class="list-item-paragraph">the equation of $CD$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q03-ii" data-list-item-path="cg-q03-ii"><div class="list-item-paragraph">the coordinates of $C$ and of $B$,<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q03-iii" data-list-item-path="cg-q03-iii"><div class="list-item-paragraph">the length of $AB$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q03-iv" data-list-item-path="cg-q03-iv"><div class="list-item-paragraph">the area of the rectangle $ABCD$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a rectangle $ABCD$ in which $A$ is $(1,\ 2)$ and $D$ is $(5,\ 0)$. The diagonal $AC$ lies on the line $y=x+1$.

Rectangle from a vertex pair and a diagonal

<p>The diagram shows the points $A(-7,\ 5)$ and $B(5,\ -4)$.</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q01-i" data-list-item-path="cg-q01-i"><div class="list-item-paragraph">the length of $AB$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q01-ii" data-list-item-path="cg-q01-ii"><div class="list-item-paragraph">the coordinates of the midpoint of $AB$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q01-iii" data-list-item-path="cg-q01-iii"><div class="list-item-paragraph">the gradient of $AB$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="cg-q01-iv" data-list-item-path="cg-q01-iv"><div class="list-item-paragraph">the equation of $AB$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows the points $A(-7,\ 5)$ and $B(5,\ -4)$.

the coordinates of the midpoint of $AB$,

Gradient of a Straight Line

Length, midpoint, gradient and equation from two points

<p>Find the equation of the line that bisects the line joining $A\left( 14,-10 \right)$ and $B\left( 3,-7 \right)$ and is parallel to the line $3x+5y-21=0$.</p>

Find the equation of the line that bisects the line joining $A\left( 14,-10 \right)$ and $B\left( 3,-7 \right)$ and is parallel to the line $3x+5y-21=0$.

<p>Find the equation of the line passing through the midpoint of $A\left( -2,5 \right)$ and $B\left( 8,-10 \right)$, and parallel to the line joining the points $C\left( 2,-7 \right)$ and $D\left( 0,-5 \right)$.</p>

Find the equation of the line passing through the midpoint of $A\left( -2,5 \right)$ and $B\left( 8,-10 \right)$, and parallel to the line joining the points $C\left( 2,-7 \right)$ and $D\left( 0,-5 \right)$.

<p>Three points have coordinates $P(-3, 4)$, $Q(7, -2)$ and $R(5, m)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $m$ if&nbsp;$PQ$ is perpendicular to $PR$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the midpoint of $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the equation of the perpendicular bisector of $PQ$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lengths of $PQ$ and $PR$,</span></div></li><li><div class="list-item-paragraph">the area of triangle $PQR$, using the results in <strong>(d)</strong>.</div></li></ol>

Three points have coordinates $P(-3, 4)$, $Q(7, -2)$ and $R(5, m)$. Find

the value of $m$ if $PQ$ is perpendicular to $PR$,

the equation of the perpendicular bisector of $PQ$,

<p>Three points have coordinates $A(1, 8)$, $B(0, 3)$ and $C(k, 5)$. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$ if&nbsp;$AC$ is perpendicular to $BC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the midpoint of $AB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the equation of the perpendicular bisector of $AB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lengths of $AB$ and $BC$,</span></div></li><li><div class="list-item-paragraph">the area of triangle $ABC$, using the results in <strong>(d)</strong>.</div></li></ol>

Three points have coordinates $A(1, 8)$, $B(0, 3)$ and $C(k, 5)$. Find

the value of $k$ if $AC$ is perpendicular to $BC$,

the equation of the perpendicular bisector of $AB$,

<p>Three of the vertices of a parallelogram $PQRS$ are $P(3,-9)$, $Q(11,2)$ and $R(-3,11)$. By finding the midpoint of $PR$, find the coordinates of $S$.</p>

Three of the vertices of a parallelogram $PQRS$ are $P(3,-9)$, $Q(11,2)$ and $R(-3,11)$. By finding the midpoint of $PR$, find the coordinates of $S$.

<p>If&nbsp;$A(1,5)$, $B(-4,8)$ and $C(-3,11)$ are three vertices of a parallelogram $ABCD$, find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the midpoint of $AC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the coordinates of $D$.</span></div></li></ol>

If $A(1,5)$, $B(-4,8)$ and $C(-3,11)$ are three vertices of a parallelogram $ABCD$, find

<p>If the coordinates of the midpoint of the line segment joining $P\left( x,y \right)$ and $Q\left( 0,-11 \right)$ are $M\left( 1,-3 \right)$, find the value of $x$ and $y$.</p>

If the coordinates of the midpoint of the line segment joining $P\left( x,y \right)$ and $Q\left( 0,-11 \right)$ are $M\left( 1,-3 \right)$, find the value of $x$ and $y$.

<p>If the coordinates of the midpoint of the line segment joining $A\left( x,y \right)$ and $B\left( -5,7 \right)$ are $M\left( -5,10 \right)$, find the value of $x$ and $y$.</p>

If the coordinates of the midpoint of the line segment joining $A\left( x,y \right)$ and $B\left( -5,7 \right)$ are $M\left( -5,10 \right)$, find the value of $x$ and $y$.

<p>Find the coordinates of the midpoint of the line segment joining</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$A(2, -5)$ and $B(10, -7)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$C\left(2, \frac{3}{7}\right)$ and $D\left(5, -\frac{2}{9}\right)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$E(3p - 2q, 3p + q)$ and $F(5p, -4q)$</span></div></li></ol>

Find the coordinates of the midpoint of the line segment joining

$C\left(2, \frac{3}{7}\right)$ and $D\left(5, -\frac{2}{9}\right)$

$E(3p - 2q, 3p + q)$ and $F(5p, -4q)$

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of the centre and the radius of the circle $x^{2} + y^{2} - 4x - 8y - 5 = 0$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the circle cuts the $x$-axis at $P$ and $Q$, find the equation of the perpendicular bisector of the line $PQ$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the line $2y - 6x + 4 = 0$ intersects the circle at $R$ and $S$, find the coordinates of $R$ and $S$. Hence show that the line segment $RS$ is a diameter of the circle.</span></div></li></ol>

Find the coordinates of the centre and the radius of the circle $x^{2} + y^{2} - 4x - 8y - 5 = 0$.

If the circle cuts the $x$-axis at $P$ and $Q$, find the equation of the perpendicular bisector of the line $PQ$.

Given that the line $2y - 6x + 4 = 0$ intersects the circle at $R$ and $S$, find the coordinates of $R$ and $S$. Hence show that the line segment $RS$ is a diameter of the circle.

Finding the Centre and Radius of a Circle

Line–Circle Intersections

<p>The point $A(2, 1)$ lies on the circle $x^2 + y^2 + kx + 4y + 7 = 0$, where&nbsp;$k$ is a constant.</p>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the centre and radius of the circle,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the coordinates of $B$ where&nbsp;$AB$ is the diameter of the circle.</span></div></li></ol>

The point $A(2, 1)$ lies on the circle $x^2 + y^2 + kx + 4y + 7 = 0$, where $k$ is a constant.

the coordinates of $B$ where $AB$ is the diameter of the circle.

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the radius and coordinates of the centre of the circle $x^2 + y^2 - 4x - 6y - 12 = 0$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$AB$ is a diameter of this circle, where&nbsp;$A$ is the point $(2, 8)$, find the coordinates of the point $B$.</span></div></li></ol>

Find the radius and coordinates of the centre of the circle $x^2 + y^2 - 4x - 6y - 12 = 0$.

Given that $AB$ is a diameter of this circle, where $A$ is the point $(2, 8)$, find the coordinates of the point $B$.

Math Question Bank