Tag: dividing a line segment into three or five equal parts

Questions Related to dividing a line segment into three or five equal parts

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

ABC is a triangle, the point P is on side BC such that $3\bar{BP}=2\bar{PC}$, the point Q is on the line $\bar{CA}$ such that $4\bar{CQ}=\bar{QA}$. If R is the common point $\bar{AP}$ & $\bar{BQ}$, then the ratio in which the fine joining CR divides $\bar{AB}$ is?

  1. $2:5$
  2. $3:8$
  3. $4:1$
  4. $6:1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using Menelaus' Theorem or vector geometry, the intersection of cevians in a triangle can be solved by setting up ratios of segments on the sides.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

If a straight line $y-x=2$ divides the region ${x}^{2}+{y}^{2}\le 4$ into two parts, then the ratio of the area of the smaller part to the area of the greater part is 

  1. $\pi-2 : 3\pi+2$
  2. $3\pi-4 : \pi+4$
  3. $\pi-3 : 3\pi+3$
  4. $3\pi-8 : \pi+8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The line divides the circle into two segments. The area of the circle is 4*pi. The line distance from the center is sqrt(2), which allows calculating the sector and triangle areas to find the ratio.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The line joining points $(3,5)$ and $(2,7)$ is divided by $X-$ axis in the ratio.

  1. $5:7$
  2. $3:2$
  3. $-5:7$
  4. $-3:2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A point on the X-axis has a y-coordinate of 0. Using the section formula, the y-coordinate is (m*y2 + n*y1) / (m+n) = 0. Solving 7m + 5n = 0 gives m/n = -5/7, indicating external division.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The point $(\dfrac{7}{4},\dfrac{7}{8})$ divides the line segment joining the points (4,-1) and (-2,4) internally in the ratio 3 : 5.

  1. True

  2. False

Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the section formula with points (4, -1) and (-2, 4) and ratio 3:5, the x-coordinate is (3*-2 + 5*4) / 8 = 14/8 = 7/4. The y-coordinate is (3*4 + 5*-1) / 8 = 7/8. The point matches.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point $M$ of $CD$. The crease will divide $BC$ in the ratio :

  1. $7:4$
  2. $5:3$
  3. $8:5$
  4. $4:1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a geometry problem involving folding a square. Let the side be 2. B is (2,2), M is (1,0). The crease is the perpendicular bisector of BM. Solving for the intersection with BC gives the ratio.

Multiple choice construction : division of a line segment dividing a line segment into three or five equal parts divsion of line segmet in given ratio constructions maths

The point which divides the line segment joining $(-2, 4), (2, 7)$ in the ratio $2:1$ externally is

  1. $(6, 10)$
  2. $(2, \dfrac{10}{3})$
  3. $(\dfrac{-4}{3}, \dfrac{2}{3})$
  4. $( \dfrac{2}{3} ,6)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For external division in ratio m:n, the formula is (mx2 - nx1)/(m-n), (my2 - ny1)/(m-n). For (2,1) and points (-2,4), (2,7): x = (2*2 - 1*-2)/(2-1) = 6. y = (2*7 - 1*4)/(2-1) = 10.