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

In $\triangle ABC, D$ is a point on AB and E is a point on BC such that DE || AC and $ar (DBE) = \dfrac {1}{2} ar (ABC)$. Find $\dfrac{AD}{AB}$

  1. $\dfrac{1 - \sqrt 2}{2}$
  2. $\dfrac{\sqrt 2 - 1}{\sqrt 2}$
  3. $\dfrac{\sqrt 2 - 1}{2}$
  4. $\dfrac{\sqrt 2 + 1}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In $\triangle ABC$

$DE \parallel BC$
$ \cfrac{AD}{BD} = \cfrac {AE}{EC} $ Basic proportionality theorm
$ AD = \cfrac{BD}{2}$
$ \cfrac {AD}{BD} = \cfrac{1}{3}$
$ \cfrac {area(DBE)}{area(ABC)} = \left (\cfrac {AD}{AB} \right)^{2} $

$ \sqrt {\cfrac {area(DBE)}{area(ABC)}} = \left (\cfrac {AD}{AB} \right) $
$ 1 - \cfrac{AD}{AB}$
$ = 1- \cfrac{1}{\sqrt {2}}$
$ = \cfrac {\sqrt{2} - 1}{\sqrt{2}}$

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

In any triangle ABC state whether following statements are true or false:
(1) the bisectors of the angles A, B, and C meet in a point,
(2) the medians, i.e. the lines joining each vertex to the middle point of the opposite side, meet in a point, and
(3) the straight lines through the middle points of the sides perpendicular to the sides meet in a point.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle bisectors meet at the incenter, the medians meet at the centroid, and the perpendicular bisectors of the sides meet at the circumcenter. All three statements are standard geometric properties.

Multiple choice maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem 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 :

  1. $17 cm$
  2. $38.4 cm$
  3. $25.6 cm$
  4. $6.4 cm$
Reveal answer Fill a bubble to check yourself
D Correct answer
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

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

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

  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The triangle formed by joining the midpoints of the sides of a triangle has an area equal to 1/4 of the area of the original triangle. (1/4) * 4 = 1.

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

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 

  1. 60

  2. 205

  3. 115

  4. 405
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $205.$

Multiple choice maths profit-loss overhead expenses introduction to total cost price of the article profit and loss with overhead expenses

An example of a production overhead would be:

  1. materials

  2. labour cost

  3. supervisory costs

  4. rent
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An example of a production overhead would be supervisory costs.
Materials and labour costs would be directly attributable to the product and would be classed as direct costs. Rent is a non production overhead but salaries of supervisors are related to production and are an overhead as they do not vary directly with output.

Multiple choice maths profit-loss overhead expenses introduction to total cost price of the article profit and loss with overhead expenses

Repairs, taxes, insurance, rent are all examples of

  1. overhead expenses

  2. interest

  3. company profit

  4. cost price
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Overhead expenses are all costs on the income statement except for direct labour, direct materials, and direct expenses. 
Overhead expenses include accounting fees, advertising, insurance, interest, legal fees, labour burden, rent, repairs, supplies, taxes, telephone bills, travel expenditures, and utilities.
Multiple choice maths profit-loss overhead expenses introduction to total cost price of the article profit and loss with overhead expenses

An ______ is the additional cost which is added to the cost price of an item.

  1. interest

  2. cost

  3. overhead

  4. profit
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By definition, Overhead is the additional cost which is added to the cost price of an item. 

Overhead expenses are all costs on the income statement except for direct labour, direct materials, and direct expenses. 
Overhead expenses include accounting fees, advertising, insurance, interest, legal fees, labour burden, rent, repairs, supplies, taxes, telephone bills, travel expenditures, and utilities.