Tag: introduction to similar triangles

Quizzes (0) Questions (73)

Questions Related to introduction to similar triangles

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
B Correct answer
Explanation

Dividing the sides of a triangle in the ratio m:n creates smaller corner triangles. The area of each corner triangle relative to the main triangle can be found using the fraction of the sides bounding the vertex, resulting in the area ratio formula (m^2 - mn + n^2) / (m + n)^2 for the inner triangle LMN.

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.