SECTION E — Long Answer Type Questions [5 Marks Each]
- 5 Marks Q1. Prove that the points $A(1, 7)$, $B(4, 2)$, $C(-1, -1)$, and $D(-4, 4)$ are the vertices of a square. Also, find the length of its diagonals and the area of the square
- 5 Marks Q2. If the points $A(6, 1)$, $B(8, 2)$, $C(9, 4)$, and $D(p, 3)$ are the vertices of a parallelogram taken in order, find the value of $p$. Hence, find the lengths of its sides and verify whether it is a rhombus or a rectangle.
- 5 Marks Q3. Find the coordinates of the circumcenter and the circumradius of the triangle whose vertices are $A(3, 0)$, $B(-1, -3)$, and $C(-1, 5)$
- 5 Marks Q4. Two opposite vertices of a square are $(-1, 2)$ and $(3, 2)$. Find the coordinates of the remaining two vertices. Also, calculate the area and perimeter of the square.
- 5 Marks Q5. If $A(-5, 7)$, $B(-4, -5)$, $C(-1, -6)$, and $D(4, 5)$ are the vertices of a quadrilateral $ABCD$, find its area by dividing it into two triangles using diagonal $AC$. Verify your result using diagonal $BD$
- 5 Marks Q6. Find the ratio in which the line segment joining the points $A(3, -3)$ and $B(-2, 7)$ is divided by the $x$-axis. Also, find the coordinates of the point of division and the length of both segments.
- 5 Marks Q7. If the vertices of a triangle are $A(1, k)$, $B(4, -3)$, and $C(-9, 7)$ and its area is $15$ square units: Find all possible values of $k$. For the positive value of $k$. Find the length of the median through vertex $A$.
- 5 Marks Q8. Prove analytically that the line segments joining the mid-points of the adjacent sides of a quadrilateral $A(-3, 2)$, $B(5, 4)$, $C(7, -6)$, and $D(-1, -8)$ form a parallelogram. Find the coordinates of the center of this parallelogram
- 5 Marks Q9. A line intersects the $x$-axis and $y$-axis at points $P$ and $Q$ respectively. If $(2, -5)$ is the mid-point of $PQ$: Find the coordinates of $P$ and $Q$. Find the length of the line segment $PQ$. Find the area of the triangle formed by $PQ$ and the coordinate axes.
- 5 Marks Q10. Let $A(4, 2)$, $B(6, 5)$, and $C(1, 4)$ be the vertices of $\triangle ABC$. Find the coordinates of point $D$ where median $AD$ meets $BC$. Verify that median $AD$ divides $\triangle ABC$ into two triangles of equal areas ($\text{area}(\triangle ABD) = \text{area}(\triangle ADC)$).
- 5 Marks Q11. Prove that the points $A(a, a)$, $B(-a, -a)$, and $C(-\sqrt{3}a, \sqrt{3}a)$ are the vertices of an equilateral triangle. Calculate its perimeter and area in terms of $a$.
- 5 Marks Q12. If $P(x, y)$ is any point on the line segment joining the points $A(a, 0)$ and $B(0, b)$, prove that $\frac{x}{a} + \frac{y}{b} = 1$. Using this relation, find the coordinates of the point that divides segment $AB$ in the ratio $2:3$ internally when $a = 5$ and $b = 10$.
- 5 Marks Q13. Find the center and radius of the circle passing through the points $A(6, -6)$, $B(3, -7)$, and $C(3, 3)$. Also, check whether the point $D(5, 1)$ lies inside, outside, or on this circle.
- 5 Marks Q14. Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts. Hence, find the sum of the distances of these division points from the origin.
- 5 Marks Q15. The three vertices of a $\triangle ABC$ are $A(2, 3)$, $B(4, -1)$, and $C(5, 2)$. Find the coordinates of the centroid $G$ of $\triangle ABC$. Verify that $GA^2 + GB^2 + GC^2 = \frac{1}{3}(AB^2 + BC^2 + CA^2) - \frac{1}{9}(AB^2 + BC^2 + CA^2)$ ... [Use direct coordinate verification of side squares and median distances]
- 5 Marks Q16. If the points $P(k-1, 2k)$, $Q(3k, 5k+1)$, and $R(k+7, 7k+5)$ are collinear: Find the value of $k$. Using the value of $k$, find the ratio in which point $Q$ divides the line segment $PR$.
- 5 Marks Q17. Find the area of the quadrilateral formed by the vertices $A(-4, -2)$, $B(-3, -5)$, $C(3, -2)$, and $D(2, 3)$. Also, find the lengths of both its diagonals $AC$ and $BD$.
- 5 Marks Q18. Prove that the coordinates of the centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are given by $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$. Use this formula to find the centroid of the triangle formed by lines intersecting the axes at $(6, 0)$, $(0, 8)$, and the origin $(0, 0)$
- 5 Marks Q19. Find the ratio in which the line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, -2)$ and $B(3, 7)$. Also, find the coordinates of the point of intersection and verify that it satisfies the given line equation.
- 5 Marks Q20. Show that the points $A(-4, -1)$, $B(-2, -4)$, $C(4, 0)$, and $D(2, 3)$ are the vertices of a rectangle. Calculate its area, perimeter, and the angle made by the diagonals at their intersection point.
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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