Tag: mid-point and its converse

Questions Related to mid-point and its converse

D,E,F are midpoints of sides BC, CA and AB of $\Delta ABC$. If perimeter of $\Delta ABC$ is 12.8 cm, then perimeter of $\Delta DEF$ is :

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. $17 cm$

  2. $38.4 cm$

  3. $25.6 cm$

  4. $6.4 cm$

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Correct Option: D
Explanation:

Given in $\triangle ABC$, $D,E,F$ are the mid points of sides $AB,BC$ and $CA$ respectively

Now using mid point theorem line segment joining the mid points of two sides is parallel to third side and also half of it.
$\therefore DF=\dfrac{1}{2}BC$
$\Rightarrow \dfrac{DF}{BC}=\dfrac{1}{2}.......(i)$
Similarly $\dfrac{DE}{AC}=\dfrac{1}{2}.........(ii)$
$\dfrac{EF}{AB}=\dfrac{1}{2}...........(iii)$
Using $(i),(ii)$ and $(iii)$
$\dfrac { DF }{ BC } =\dfrac { DE }{ AC } \dfrac { EF }{ AB } =\dfrac { 1 }{ 2 } $
$\dfrac { DF }{ BC } =\dfrac { DE }{ AC } \dfrac { EF }{ AB } =\dfrac { 1 }{ 2 } \ \therefore \triangle ABC\sim \triangle DEF$
Now if triangles are similar then ratio of their perimeter is equal to ratio of  of their corresponding sides.
$\ \dfrac { Perimeter(\triangle DEF) }{ Perimeeter(\triangle ABC) } =\dfrac { 1 }{ 2 } $
$\Rightarrow $ Perimeter of $\triangle ABC=\dfrac{1}{2}\times 12.8=6.4$ cm

If A, B and C are the midpoint of the sides PQ, QR and PR of $\triangle $PQR respectively, then the area of $\triangle $ABC equals if area of $\triangle PQR$ is $4$ units

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. $1$

  2. $2$

  3. $3$

  4. $4$

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Correct Option: A

The sides $AB, BC$ and $CA$ of a triangle $ABC$ have $3, 4$ and $5$ interior points respectively on them.The number of triangles that can be constructed using these interior points as vertices is 

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. 60

  2. 205

  3. 115

  4. 405

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Correct Option: B
Explanation:

No. of ways $={^{ 3 }{ { C } _{ 1 } }}\times {^{ 4 }{ { C } _{ 1 }} }\times {^{ 5 }{ { C } _{ 1 } }}+{^{ 3 }{ { C } _{ 2 } }}\left( ^{ 4 }{ { C } _{ 1 } }+{^{ 5 }{ { C } _{ 1 } }}\right)+{^{ 4 }{ { C } _{ 2 }} } \left(^{ 3 }{ { C } _{ 1 } }+{^{ 5 }{ { C } _{ 1 }} }\right) +{^{ 5 }{ { C } _{ 2 }} }\left( ^{ 3 }{ { C } _{ 1 } }+{^{ 4 }{ { C } _{ 1 } }}\right)$

$\Rightarrow$ No of ways $=60+3\left( 4+5\right) +6\left( 3+5\right) +10 \left( 3+4\right)$
$\Rightarrow$ No of ways $=60+27+48+70=205.$
Hence, the answer is $205.$

In $\triangle ABC, D$ and $E$ are the mid point of $\bar {BC}$ and $\bar {AC}$ respectively. $\bar {AD}$ and $\bar {BE}$ intersect each other in $G.A$ line $m$ passing through $D$ and parallel to $\overleftrightarrow { BE } $ intersects $\bar {AC}$ in $K$.
then $AC=4CK$

maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

  1. True

  2. False

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Correct Option: A

State true or false:

$ D, E $ and $ F $ are the mid-points of the sides $ AB, BC $ and  $ CA $ of an isosceles $ \bigtriangleup ABC $  in which $ AB= BC $. then
 $ \bigtriangleup DEF $  is also isosceles.

maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

  1. True

  2. False

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Correct Option: A
Explanation:

AB = AC
Hence, $\angle ABC = \angle ACB$ (Isosceles triangle property)
Now, since, D and F are mid point of AB and AC respectively, thus DF II BC (Mid point theorem)
Hence, 
$\angle ADF = \angle ABC$ and $\angle AFD = \angle ACB$ (Corresponding angles)
Thus, $\angle ADF = \angle ABC = \angle AFD = \angle ACB$ 
Now, In $\triangle$ ADF and FEC
$\angle ADF = \angle FEC$ (Corresponding angles of parallel lines EF and AB)
$\angle AFD = \angle ACB $(Corresponding angles of parallel lines DF and BC)
AF = FC (F is the mid point of AC)
Thus $\triangle ADF \cong \triangle FEC$ (AAS rule)
Hence, AD = FE (corresponding sides of congruent triangles)
Similarly, we can prove, AF = DE
Since, AD = AF (half lengths of equal sides, AB and AC)
Thus, EF = DE or $\triangle$ DEF is an isosceles triangle.

In $\Delta ABC$, D and E are mid points of AB and BC respectively and $\angle ABC=90^o$, then

maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

  1. $AE^2+CD^2=AC^2$

  2. $AE^2+CD^2=\frac {5}{4}AC^2$

  3. $AE^2+CD^2=\frac {3}{4}AC^2$

  4. $AE^2+CD^2=\frac {4}{5}AC^2$

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Correct Option: B

Find the midpoint of the segment connecting the points $(a, -b)$ and $(5a, 7b)$.

maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

  1. $(3a, -3b)$

  2. $(2a, -3b)$

  3. $(3a, -4b)$

  4. $(-2a, 4b)$

  5. none of these

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Correct Option: D
Explanation:

The mid point is $(\dfrac{a-5a}2,\dfrac{b+7b}2)$.i.e $(-2a,4b)$
Option D is correct.

Fill in the blanks:
(i) The ling segment joining a vertex of a triangle to the midpoint of its opposite side is called a $\underline { P } $ of the triangle.
(ii) The perpendicular line segment from a vertex of a triangle to its opposite is called an $\underline { Q } $ of the triangle
(iii) A triangle has $\underline { R } $ altitudes and $\underline { S } $ medians

maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

  1. $P-$ Altitude; $Q$- Median; $R-1$; $S-1$

  2. $P-$ Altitude; $Q$- Median; $R-3$; $S-3$

  3. $P-$ Median; $Q$- Altitude; $R-3$; $S-3$

  4. $P-$ Median; $Q$- Altitude; $R-2$; $S-3$

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Correct Option: C
Explanation:

Medians : The line segment from any vertex of a triangle to the midpoint of its opposite side is called medians of triangle. 

Altitude : The perpendicular drawn from a vertex to opposite side is called as altitude. 
A triangle has $3$ altitudes & $3$ medians.

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the _______ side.

maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

  1. first

  2. second

  3. third

  4. none of the above

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Correct Option: C
Explanation:

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Fill in the blank:

Line joining the mid-points of any two sides of a triangle is _____ to the third side.

maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

  1. perpendicular

  2. parallel

  3. Inclined at $60^o$

  4. None of these

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Correct Option: B
Explanation:

Line joining the mid-points of any two sides of a triangle is parallel to the third side. 

This is the standard Midpoint theorem of a triangle.