2 Marks Questions

Question 12 Marks

Find the distance between the points:
$A(1, -3)$ and $B(4, -6)$

Answer

The given points are $A(1, -3)$ and $B(4, -6)$
Then, $(x_1 = 1, y_1 = -3)$ and ($x_2 = 4 $and $y_2 = -6$)
$\therefore\text{AB}=\sqrt{(\text{x}_2-\text{x}_1)^2+(\text{y}_2-\text{y}_1)^2}$
$=\sqrt{(\text{4}-\text{1})^2+(-\text{6}+\text{3})^2}$
$=\sqrt{(\text{3})^2+(-\text{3})^2}$
$=\sqrt{\text{9}+\text{9}}$
$=\sqrt{\text{18}}=3\sqrt{2}\text{ units}.$

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Question 22 Marks

If the coordinates of points A and B are (-2, -2) and (2, -4) respectively, find the coordinates of the points P such that $\text{AP}=\frac{3}{7}\text{AB},$ where P lies on the line segment AB.

Answer

Here, P(x, y) divides line segment AB, such that
$\text{AP}=\frac{3}{7}\text{AB}$
$\Rightarrow\frac{\text{AP}}{\text{AB}}=\frac{3}{7}$
$\Rightarrow\frac{\text{AB}}{\text{AP}}=\frac{7}{3}$
$\Rightarrow\frac{\text{AB}}{\text{AP}}-1=\frac{7}{3}-1$
$\Rightarrow\frac{\text{AB}-\text{AP}}{\text{AP}}=\frac{7-3}{3}$
$\Rightarrow\frac{\text{BP}}{\text{AP}}=\frac{4}{3}$
$\Rightarrow\frac{\text{AP}}{\text{BP}}=\frac{3}{4}$
$\therefore$ P divides AB in the ratio 3 : 4
$\text{x}=\frac{3\times2+4(-2)}{3+4};\text{y}=\frac{3\times(-4)+4(2)}{3+4}$
$\text{x}=\frac{6-8}{7};\text{y}=\frac{-12-8}{7}$
$\text{x}=\frac{-2}{7};\text{y}=\frac{-20}{7}$
$\therefore$ The coordinate of P are $\Big(\frac{-2}{7},\frac{-20}{7}\Big).$

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Question 32 Marks

Find the coordinate of the points of trisection of the line segment joining the points A(7, -2) and B(1, -5).

Answer

We have given that the coordinates of the Points of trisection of the line segment joining the points A(7, -2) and B(1, -5)
Therefore, the line AB divides sectionally into two parts C and D into 1 : 2 and 2 : 1
Now, we have use the Section Formula,
For 1 : 2 $= \text{ma}+\frac{\text{nb}}{\text{a+b}}, \text{ma}+\frac{\text{nb}}{\text{a+b}} $
$\Rightarrow1\times1+2\times\frac{7}{1+2}, 1\times(-5)+2\times(-2)\frac{7}{1+2})$
$ = \frac{15}{3}, \frac{-9}{3} = (5, -3)$
Similarly, For the Ratio 2 : 1
For 1 : 2 $\Rightarrow\frac{\text{ma+nb}}{\text{a+b}},\frac{\text{ma+nb}}{\text{a+b}}$$=2\times1+1\times\frac{7}{2}+1, 2\times(-5)+1\times\frac{(-2)}{2+1} -4)$
$= \frac{9}{3}, \frac{-12}{3} = (3, -4)$
Hence, the coordinates of the point of the trisection of the line segments are C(5, -3) and D(3, -4)

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Question 42 Marks

Find the area of $\triangle\text{ABC}$ whose vertices are:
$A(3, 8), B(-4, 2)$ and $C(5, -1)$

Answer

$A(3, 8), B(-4, 2)$ and $C(5, -1)$ are the vertices of $\triangle\text{ABC}.$
Then, $(x_1 = 3, y_1= 8), (x_2 = -4, y_2 = 2)$ and $(x_3 = 5, y_3 = -1)$
Area of triangle ABC
$=\frac{1}{2}\big\{\text{x}_1(\text{y}_2-\text{y}_3)+\text{x}_2(\text{y}_3-\text{y}_1)+\text{x}_3(\text{y}_1-\text{y}_2)\big\}$
$=\frac{1}{2}\big\{3(2-(-1))+(-4)(-1-8)+5(8-2)\big\}$
$=\frac{1}{2}\big\{3(2+1)-4(-9)+5(6)\big\}$
$=\frac{1}{2}\big\{9+36+30\big\}$
$=\frac{1}{2}(75)$
$=37.5\ \text{sq}.\text{units}$

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Question 52 Marks

Find the distance between the points:
$A(-6, -4)$ and $B(9, -12)$

Answer

The given points are $A(-6, -4)$ and $B(9, -12)$
Then,$(x_1 = -6, y_1 = -4)$ and ($x_2 = 9$ and $y_2 = -12$)
$\therefore\text{AB}=\sqrt{(\text{x}_2-\text{x}_1)^2+(\text{y}_2-\text{y}_1)^2}$
$=\sqrt{(\text{6}+\text{9})^2+(-\text{12}+\text{4})^2}$
$=\sqrt{(\text{15})^2+(-\text{8})^2}$
$=\sqrt{\text{225}+\text{64}}$
$=\sqrt{\text{289}}=17\text{ units}.$

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Question 62 Marks

Find the distance between the points:
$A(9, 3)$ and $B(15, 11)$

Answer

The given points are$ A(9, 3)$ and $B(15, 11)$
Then, $(x_1 = 9, y_1 = 3)$ and ($x_2 = 15$ and $y_2 = 11$)
$\therefore\text{AB}=\sqrt{(\text{x}_2-\text{x}_1)^2+(\text{y}_2-\text{y}_1)^2}$
$=\sqrt{(\text{15}-\text{9})^2+(\text{11}-\text{3})^2}$
$=\sqrt{\text{6}^2+\text{8}^2}$
$=\sqrt{\text{36}+\text{64}}$
$=\sqrt{\text{100}}=10\text{ units}.$

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Question 72 Marks

Find the distance between the points:
$A(7, -4)$ and $B(-5, 1)$

Answer

The given points are $A(7, 4)$ and $B(-5, 1)$
Then, $(x_1 = 7, y_1 = -4)$ and ($x_2 = -5$ and $y_2 = 1$)
$\therefore\text{AB}=\sqrt{(\text{x}_2-\text{x}_1)^2+(\text{y}_2-\text{y}_1)^2}$
$=\sqrt{(-\text{5}-\text{7})^2+(\text{1}+\text{4})^2}$
$=\sqrt{(-\text{12})^2+(\text{5})^2}$
$=\sqrt{\text{144}+\text{25}}$
$=\sqrt{\text{169}}=13\text{ units}.$

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