Class 10 (All in one) – Coordinate GeometryContact Number: 9667591930 / 8527521718

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Find the coordinates of the points which divide the line segment joining A(–2, –2) and B(2, 8) into four equal parts.

Find the lengths of the medians of the triangle whose vertices are (5, 6), (3, 8) and (–1, 2).

The points A, B and C are collinear and AB = BC. If the coordinates of A, B and Care (3, a), (1, 3) and (b, 4) respectively, then find the values of a and b. **CBSE 2013**

Find the coordinates of the point Q on the X-axis, which lies on the perpendicular bisector of the line segment joining the points A(–5, –2) and B (4, –2). Name the type of triangle formed by the points Q, A and B. **NCERT Exemplar**

Two vertices of a triangle are (8, –6) and (–4, 6). The area of the triangle is 120 sq. units. Find the third vertex if it lies on x – 2y = 6.

If centroid of triangle is (3, –5) and two vertices of triangle are (4, –8) and (3, 6), then find the third vertex.

In a ∆ABC, the coordinates of points A, B and C are (3, 2), (6, 4) and (9, 3), respectively. Find the coordinates of centroid G. Also, find the areas of ∆ABG and ∆ACD. **CBSE 2013**

If the points A (1, –2), B (2, 3), C(–3, 2) and D(–4, –3) are the vertices of a parallelogram ABCD, then taking AB as the base, find the height of this parallelogram. **CBSE 2013**

Find the area of a parallelogram ABCD, if three of its vertices are A(2, 4), B(2 + *, 5) and C(2, 6). **CBSE 2013**

Find the perpendicular distance of *A* (5, 12) from the *Y*-axis.

If *A* is a point on *Y*-axis, whose ordinate is 3 and coordinates of point *B* is (‒5, 2), then find the distance *AB*.

If $(3,\frac{3}{4})$ is the mid-point of the line segment joining the points (*K*, 0) and $(7,\frac{3}{2})$, then find the value of *k*. **CBSE 2015**

Find the radius of the circle whose end points of diameter are (24, 1) and (2, 23). **CBSE 2015**

What is the distance between the points $(10\text{cos}30\xb0,0)$ and $(0,10\text{cos6}0\xb0)$?

Find the point which lies on the perpendicular bisector of the line segment joining the points *A*(‒2, ‒5) and *B *(2, 5).** NCERT Exemplar**

Find the area of triangle with vertices (*a*, *b* + *c*), (*b*, *c* + *a*) and (*c*, *a* + *b*). **NCERT Exemplar**

*AOBC* is a rectangle whose three vertices are *A* (0, 3), *O* (0, 0) and *B* (5, 0). What are the length of its diagonals? **NCERT Exemplar**

Find the perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0). **NCERT Exemplar**

[**Hint **The perimeter of a triangle is the sum of lengths of its three sides, so first find the length of the three sides and then add them.]

If the point* P* (2, 1) lies on the line segment joining the points *A* (4, 2) and *B* (8, 4), then find the relation between *AP* and *AB*. **NCERT Exemplar**

If the point *C*(*K*, 4) divides the join of points *A*(2, 6) and *B*(5, 1) in the ratio 2 : 3, find the value of *k*?

In the following figure, find the area of Δ*ABC*. **CBSE 2013**

If the centroid of triangle formed by points *P* (*a*, *b*), *Q *(*b*, *c*) and *R *(*c*, *a*) is at the origin. What is the value of (*a* +* b* + *c*)?

Prove that the points (3, 0), (6, 4) and (‒1, 3) are the vertices of a right angled isosceles triangle.** CBSE 2016**

Check, whether the points (‒4, 0), (4, 0) and (0, 3) are the vertices of an isosceles triangle or equilateral triangle. **NCERT Exemplar, CBSE 2014**

If the points *A* (4, 3) and B (*x*, 5) are on the circle with center *O* (2, 3), then find the value of *x*^{2} + 5.

Show that if a circle has its centre at the origin and a point *P*(5, 0) lies on it, then the point *Q* (6, 8) lies outside the circle. **NCERT Exemplar**

Point *P* divides the line segment joining the points *A* (‒1, 3) and *B* (9, 8) such that $\frac{AP}{BP}=\frac{k}{1}$. If *P* lies on the line *x* ‒ *y* + 2 = 0, then find the value of *k*. **CBSE 2010**

Show that the points *A* (‒ 6, 10), *B* (‒ 4, 6) and *C* (3, ‒ 8) are collinear, such that $AB=\frac{2}{9}AC$. **NCERT Exemplar**

If the points (*P*, *q*), (*M*, *n*) and (*P* ‒ *M*, *q* ‒ *n*) are collinear. Show that *Pn* = *qM*. **CBSE 2010**

Prove that the area of triangle whose vertices are (*t*, *t* ‒ 2), (*t* + 2, *t* + 2) and (*t* + 3, *t*), is independent of *t*.

Show that Δ*ABC *with vertices *A* (‒ 2, 0), *B* (0, 2) and *C* (2, 0) is similar to Δ*DFE *with vertices *D* (‒ 4, 0), *E* (4, 0) and F (0, 4). **NCERT Exemplar**

If the point *A* (2, ‒ 4) is equidistant from *P* (3, 8) and *Q* (‒ 10, *y*), then find the value of *y*. Also, find distance *PQ*. **NCERT Exemplar**

Show that quadrilateral *PQRS* formed by vertices *P* (22, 5), *Q* (7, 10), *R* (12, 11) and *S* (3, 24) is not a parallelogram. **CBSE 2015**

*ABCD* is a rectangle formed by the points *A* (‒1, ‒1), *B* (‒1, 4), *C* (5, 4) and *D* (5, ‒1). *P*, *Q*, *R* and *S* are the mid-points of *AB*, *BC*, *CD*, and *DA*, respectively. Is the quadrilateral *PQRS* a square, a rectangle or a rhombus? Justify your answer.

If the area of Δ*ABC* formed by *A*(*x*, *y*), *B*(1, 2) and *C*(2, 1) is 6 sq units, then prove that *x* + *y* = 15 or *x* + *y* + 9 = 0. **CBSE 2013**

In the given figure, in Δ*ABC*, *D* and *E* are the mid-points of the sides *BC* and *AC* respectively. Find the length of *DE*. Prove that $DE=\frac{1}{2}AB$. **CBSE 2011**** **

Find the area of parallelogram *ABCD*, if three of its vertices are *A*(2, 4), *B*$(2+\sqrt{3},\text{}5)$ and *C* (2, 6). **CBSE 2013**

If *a* ≠ *b* ≠ *c*, then prove that the points (*a*, *a*^{2}), (*b*, *b*^{2}) and (*c*, *c*^{2}) can never be collinear.

Four points *A* (6, 3), *B* (‒3, 5) *C* (4, ‒2) and *D* (*x*, 3*x*) are given such that $\frac{\Delta DBC}{\Delta ABC}=\frac{1}{2}$, find *x*.

The vertices of a Δ*ABC* are *A* (7, 8), *B* (4, 2) and *C* (8, 2). The mid-point of the side *BC* is (6, 2). Show that the median *AD* divides the Δ*ABC* into two triangles equal in area. Also, find the area of Δ*ABC* to verify your answer. **CBSE 2015**

The vertices of Δ*ABC *are *A* (‒2, 0), *B* (2, 0) and *C* (0, 2) and that of Δ*PQR* are *P* (‒4, 0), *Q* (4, 0) and *R* (0, 4). Verify that the ratio of the areas of the two triangles is equal to the square of the ratio of their corresponding sides. **CBSE 2015**

*A* (6, 1), *B* (8, 2) and *C* (9, 4) are three vertices of a parallelogram *ABCD*. If *E* is the mid-point of *DC*, then find the area of Δ*ADE*. **NCERT Exemplar**

The coordinates of *A*, *B*, *C* are (7, 4), (‒2, 6) and (5, ‒1), respectively and *P* is any point (*x*, *y*). Show that the ratio of the areas of Δ*PBC* and Δ*ABC* is $\left|\frac{x+y-4}{7}\right|$.

If the points *A*(1, ‒2), *B*(2, 3), *C*(‒3, 2) and *D*(‒ 4, ‒ 3) are the vertices of parallelogram *ABCD*, then taking *AB* as the base, find the height of the parallelogram. **CBSE 2013**

Observe the graph given below and state whether Δ*ABC* is scalene, isosceles or equilateral. Justify your answer. Also, find the area of Δ*ABC*.

To raise social awareness about hazards of smoking, a school decided to start “NO SMOKING” campaign.10 students are asked to prepare campaign banners in the shape of triangle (as shown in the figure).

(i) If cost of 1cm^{2} of banner is ₹ 2, then find the overall cost incurred on such campaign.

(ii) Which mathematical concept is used in this question?

(iii) Which value is depicted in this question?

In a sports day celebration, Pushpraj and Shani are standing at positions *A* and *B* whose coordinates are (2, ‒ 2) and (4, 8), respectively. The teacher asked Deepanshi to fix the country flag at the mid-point of the line joining points *A* and *B*.

(i) Find the coordinates of the mid-point.

(ii) Which mathematical concept is used to solve the question?

(iii) What type of value is depicted here?

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