Mid-Point Theorem and Its Converse [Including Intercept Theorem] - Q&A and Flashcards

Question

According to the Mid-point Theorem, the line segment joining the mid-points of any two sides of a triangle is _____ to the third side.

Answer

parallel

Question

The Mid-point Theorem states that the line segment joining the mid-points of two sides of a triangle is equal to _____ of the third side.

Answer

half

Question

In the proof of the Mid-point Theorem for $\triangle ABC$, what specific construction is made to prove $DE \parallel BC$?

Answer

Draw $CF$ parallel to $BA$ to meet $DE$ produced at $F$.

Question

Which congruence criterion is used to prove $\triangle ADE \cong \triangle CFE$ in the Mid-point Theorem proof?

Answer

$A.S.A.$ (Angle-Side-Angle)

Question

In the proof of Theorem 6, why is $\angle AED$ equal to $\angle CEF$?

Answer

They are vertically opposite angles.

Question

In the Mid-point Theorem proof, what property of quadrilateral $BCFD$ is used to conclude that $DF = BC$?

Answer

It is a parallelogram.

Question

State the Converse of the Mid-point Theorem regarding a line drawn through the mid-point of one side of a triangle parallel to another.

Answer

The line bisects the third side.

Question

To prove the Converse of the Mid-point Theorem, what shape is $BCFD$ constructed to be?

Answer

A parallelogram

Question

What figure is obtained by joining the mid-points of the adjacent sides of any quadrilateral?

Answer

A parallelogram

Question

In $\triangle ABC$, if $P$ and $Q$ are mid-points of $AB$ and $BC$, then $PQ$ is parallel to which segment?

Answer

$AC$

Question

If $P, Q, R, S$ are mid-points of sides $AB, BC, CD, DA$ of a quadrilateral, why is $PQ \parallel SR$?

Answer

Both are parallel to diagonal $AC$.

Question

In a right-angled triangle $ABC$ with $\angle B = 90^{\circ}$, if $D$ is the mid-point of hypotenuse $AC$, then $BD$ is equal to _____.

Answer

$\frac{1}{2} AC$

Question

In the proof that the median to the hypotenuse of a right triangle is half the hypotenuse, which congruence rule proves $\triangle AED \cong \triangle BED$?

Answer

$S.A.S.$ (Side-Angle-Side)

Question

In a trapezium, the line segment joining the mid-points of the non-parallel sides is parallel to the parallel sides and equal to _____.

Answer

half the sum of the lengths of the parallel sides

Question

Formula: If $AB \parallel DC$ in a trapezium and $E, F$ are mid-points of $AD$ and $BC$, then $2EF = $ _____.

Answer

$AB + DC$

Question

The line segment joining the mid-points of the diagonals of a trapezium is equal to half the _____ between the parallel sides.

Answer

difference

Question

Formula: If $E$ and $F$ are mid-points of diagonals $AC$ and $BD$ in trapezium $ABCD$ ($AB \parallel DC$), then $EF = $ _____.

Answer

$\frac{1}{2}(AB - DC)$

Question

The Intercept Theorem states that if a transversal makes equal intercepts on three or more parallel lines, then _____.

Answer

any other transversal will also make equal intercepts

Question

In the Intercept Theorem ($l \parallel m \parallel n$), if transversal $AB$ provides $PQ = QR$, what is the relationship between intercepts $LM$ and $MN$ on transversal $CD$?

Answer

$LM = MN$

Question

In the proof of the Intercept Theorem, which two triangles are proved congruent to show $PS = QT$?

Answer

$\triangle PQS$ and $\triangle QRT$

Question

The quadrilateral formed by joining the mid-points of the sides of a rectangle is a _____.

Answer

rhombus

Question

The quadrilateral formed by joining the mid-points of the sides of a rhombus is a _____.

Answer

rectangle

Question

The quadrilateral formed by joining the mid-points of the sides of a square is a _____.

Answer

square

Question

In $\triangle ABC$, if medians $BE$ and $CF$ are produced to $M$ and $N$ such that $BE = EM$ and $CF = FN$, then $A$ is the _____ of $MN$.

Answer

mid-point

Question

In $\triangle ABC$, if medians $BE$ and $CF$ are produced to $M$ and $N$ such that $BE = EM$ and $CF = FN$, then $NAM$ is a _____.

Answer

straight line

Question

In parallelogram $ABCD$, if $P$ and $Q$ are mid-points of $AB$ and $CD$ respectively, the lines $DP$ and $BQ$ trisect which diagonal?

Answer

$AC$

Question

In $\triangle ABC$, if $D, E, F$ are mid-points of sides $BC, CA, AB$, and $AB = AC$, then $\triangle DEF$ is an _____ triangle.

Answer

isosceles

Question

In a trapezium $ABCD$ where $AB \parallel DC$, if $P$ is the mid-point of $AD$ and a line through $P$ parallel to $AB$ meets $BC$ at $R$, then $PR = $ _____.

Answer

$\frac{1}{2}(AB + CD)$

Question

If the diagonals of a quadrilateral intersect at right angles, joining the mid-points of its adjacent sides forms a _____.

Answer

rectangle

Question

If $l \parallel m \parallel n$ and transversal $p$ cuts them at $A, B, C$ such that $AB = BC$, then any transversal $q$ cutting them at $D, E, F$ will satisfy _____.

Answer

$DE = EF$

Question

In $\triangle ABC$, if $D$ and $E$ are mid-points of $AB$ and $AC$, then the perimeter of $\triangle ADE$ is _____ the perimeter of $\triangle ABC$.

Answer

half

Question

In a trapezium $ABCD$ with $AB \parallel DC$, if $E$ and $F$ are mid-points of non-parallel sides $AD$ and $BC$, then $EF$ is _____ to $AB$.

Answer

parallel

Question

If $P, Q, R$ are mid-points of sides $BC, CA, AB$ of $\triangle ABC$, then $\triangle PQR$ is _____ to the other three triangles formed.

Answer

congruent

Question

In $\triangle ABC$, if $M$ is the mid-point of $AB$ and a line through $M$ parallel to $BC$ cuts $AC$ at $N$, what is the length of $AN$ if $AC = 10\text{ cm}$?

Answer

$5\text{ cm}$

Question

If $ABCD$ is a rhombus and $P, Q, R, S$ are mid-points of sides $AB, BC, CD, DA$, what is the measure of each interior angle of $PQRS$?

Answer

$90^{\circ}$

Question

In $\triangle ABC$, if $D$ and $E$ are mid-points of $AB$ and $AC$, then the quadrilateral $BCED$ is a _____.

Answer

trapezium

Question

In a trapezium, the segment joining mid-points of diagonals is parallel to the _____ sides.

Answer

parallel

Question

What property of mid-points ensures that the four triangles formed by joining mid-points of a triangle's sides are all congruent?

Answer

The Mid-point Theorem

Question

In $\triangle ABC$, if $D$ is the mid-point of $AB$, $E$ is the mid-point of $AC$, and $BC = 14\text{ cm}$, find the length of $DE$.

Answer

$7\text{ cm}$

Question

If a line segment joining mid-points of two sides of a triangle is $5\text{ cm}$ long, what is the length of the third side?

Answer

$10\text{ cm}$

Question

Cloze: The Converse of the Mid-point Theorem is essentially a special case of the _____ Theorem.

Answer

Intercept

Question

Concept: Intercept

Answer

Definition: The length of a segment of a transversal cut by two parallel lines.

Question

In a triangle, how many line segments can be drawn that are both parallel to a side and half its length?

Answer

Three

Question

If $P, Q, R$ are mid-points of the sides of $\triangle ABC$, then $\triangle PQR$ is called the _____ triangle of $\triangle ABC$.

Answer

medial

Question

In the proof of the Intercept Theorem, the quadrilaterals $PSML$ and $QTNM$ are both _____.

Answer

parallelograms

Question

If the mid-points of the sides of a quadrilateral form a square, what must be true about the diagonals of the original quadrilateral?

Answer

They are equal and perpendicular.

Question

If $D$ and $E$ are mid-points of $AB$ and $AC$ in $\triangle ABC$, then $\triangle ADE$ is _____ to $\triangle ABC$ in a $1:2$ ratio of sides.

Answer

similar

Question

In a right-angled triangle, the distance from the right-angle vertex to the mid-point of the hypotenuse is equal to the _____ of the hypotenuse.

Answer

radius (or half length)

Question

If $E$ is the mid-point of side $AD$ of trapezium $ABCD$ ($AB \parallel DC$) and $EF \parallel AB$ meets $BC$ at $F$, then $F$ is the _____.

Answer

mid-point of $BC$

Question

If a triangle has sides of $6\text{ cm}$, $8\text{ cm}$, and $10\text{ cm}$, what is the perimeter of the triangle formed by its mid-points?

Answer

$12\text{ cm}$

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