Tag: introduction to shapes

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Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

The sides of a triangle are $5$ cm, $6$ cm and $7$ cm. One more triangle is formed by joining the midpoints of the sides. The perimeter of the second triangle is:

  1. $18$ cm
  2. $12$ cm
  3. $9$ cm
  4. $6$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the $\triangle ABC $ have sides $AB = 5$cm,

$BC = 6$ cm and $AC = 7$cm.
Let the midpoints of the sides AB and AC be points D and E respectively.
$\therefore \dfrac {AD}{DB} = \dfrac {AE}{EC}$         ...By B.P.T

$\therefore \dfrac {AD+DB}{DB} = \dfrac {AE+EC}{EC}$   ....By Componendo

$\therefore \dfrac {AB}{DB} = \dfrac {AC}{EC}$      ......(1)

Also, $\angle BAC \cong \angle DAE$    ....(2)

$\therefore \triangle ABC \sim \triangle ADE$      ....SAS test of similarity

$\therefore \dfrac {AB}{AD} = \dfrac {BC}{DE} = \dfrac {AC}{AE}$       ....C.S.S.T

But $\dfrac {AB}{AD} = \dfrac {AB}{\frac 12 AB} = \dfrac 12$

$\therefore \dfrac {BC}{DE} = \dfrac 12$


Perimeter $(\triangle ADE) = AD + DE + AE$ 
$ = \dfrac 12 AB + \dfrac 12 BC + \dfrac 12 AC$

$= \dfrac 12 \left(AB + BC + AC \right)$

$ = \dfrac 12 \times 18 = 9$ cm.

So, option C is correct.

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

Point L, M and N lie on the sides AB, BC and CA of the triangle ABC such that $\ell (AL) : \ell (LB) = \ell (BM) : \ell (MC) = \ell (CN) : \ell (NA) = m : n$, then the areas of the triangles LMN and ABC are in the ratio

  1. $\dfrac{m^2}{n^2}$
  2. $\dfrac{m^2 - mn + n^2}{(m + n)^2}$
  3. $\dfrac{m^2 - n^2}{m^2 + n^2}$
  4. $\dfrac{m^2 + n^2}{(m + n)^2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

A man of height 1.8 metre is moving away from a lamp post at the  rate of 1.2 m/sec . If the height of the lamp post be 4.5 metre , then the rate at which the shadow of the  man is lengthening is 

  1. $0.4 m/sec$
  2. $0.8m/sec$
  3. $1.2m/sec$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let x be the distance of the man from the post and y be the length of the shadow. By similar triangles, y/1.8 = (x+y)/4.5. Simplifying gives 4.5y = 1.8x + 1.8y, so 2.7y = 1.8x, or y = (2/3)x. The rate of change dy/dt = (2/3) * dx/dt = (2/3) * 1.2 = 0.8 m/sec.