Distance and Midpoint Formula Worksheet

**Page 3**

21.

One endpoint of a line-segment is (5$x$, 7$y$) and other endpoint is (2, 4). Find the value of $x$ and $y$, if the midpoint of the segment is (3, 5).

a. | $x$ = $\frac{4}{5}$ and $y$ = $\frac{7}{8}$ | ||

b. | $x$ = $\frac{4}{5}$ and $y$ = $\frac{6}{7}$ | ||

c. | $x$ = $\frac{5}{6}$ and $y$ = $\frac{6}{7}$ | ||

d. | $x$ = 5 and $y$ = 7 |

The midpoint of the segment is (3, 5).

Midpoint of a line-segment with endpoints (

[Use the midpoint formula.]

Midpoint = (

[Replace (

(

[Equate the midpoints.]

[Equate

5

5

[Subtract 2 from both sides.]

[Divide by 5 on both sides.]

[Equate

7

7

[Subtract 4 from both sides.]

[Divide by 7 on both sides.]

Correct answer : (2)

22.

What is the midpoint of the line-segment with one endpoint at origin and other endpoint at ($x$_{1}, $y$_{1})?

a. | ($x$ _{1}, $y$_{1}) | ||

b. | ($\frac{{x}_{1}}{2}$, $\frac{{y}_{1}}{2}$) | ||

c. | (2$x$ _{1}, $y$_{1}) | ||

d. | (${x}_{1}$ - 2, ${y}_{1}$ - 2) |

Midpoint of a line-segment with endpoints (

[Use the midpoint formula.]

Midpoint = (

[Replace (

= (

[Simplify the numerators.]

The midpoint of the line-segment with one endpoint at origin and other endpoint at (

Correct answer : (2)

23.

Find the midpoint of the segment OB if B is at (8, 10) and O is the origin.

a. | (4, - 5) | ||

b. | (- 4, 5) | ||

c. | (- 4, - 5) | ||

d. | (4, 5) |

Midpoint of a line segment with endpoints (0, 0) and (

[Use the midpoint formula.]

Midpoint of OB = (

[Replace (

= (4, 5)

[Write the fractions in simplest form.]

The midpoint of the line-segment OB = (4, 5).

Correct answer : (4)

24.

In the line-segment AB, the coordinates of A are ($x$, $y$) and B are (4$x$, 5$y$) and the midpoint of the segment AB is at (15, 6). Find the coordinates of A and B.

a. | A(- 6, 2) and B(- 24, 10) | ||

b. | A(6, - 2) and B(24, - 10) | ||

c. | A(- 6, - 2) and B(- 24, - 10) | ||

d. | A(6, 2) and B(24, 10) |

Midpoint of a line-segment with endpoints (

[Use the midpoint formula.]

Midpoint of AB = (

[Replace (

= (

[Simplify the numerators.]

(

[Equate the midpoints.]

[Equate

[Simplify.]

The coordinates of A are (

The coordinates of B are (4

Correct answer : (4)

25.

The coordinates of A are (5$p$, 6$p$) and its distance from origin is 5$\sqrt{61}$units. Find the values of $p$.

a. | 25 and - 25 | ||

b. | 11 and - 11 | ||

c. | 5 and - 5 | ||

d. | 6 and - 6 |

Distance of a point (

[Write the distance formula.]

Distance of A from origin =

[Replace (

=

[Evaluate powers.]

=

[add.]

The distance of point A from origin is 5

[Equate distances.]

61

[Squaring on both sides.]

[Divide each side by 61.]

= ± 5

[Find the square root.]

The values of

Correct answer : (3)

26.

The coordinate plane shows a parallelogram ABCD. Find the length of the diagonal AC.

a. | 4.5 units | ||

b. | 8.6 units | ||

c. | 6.4 units | ||

d. | 5.8 units |

Distance between the line segment with endpoints (

Distance between A and C =

[Replace (

=

[Subtract.]

=

[Evaluate powers.]

=

[Add.]

= 8.6

Length of the line segment = Distance between the endpoints of the segment.

The length of the diagonal AC is 8.6 units.

Correct answer : (2)

27.

The coordinates of three points are A(12, 0), B(24, 0) and C(12, 9). What kind of figure is ABC?

a. | scalene triangle | ||

b. | equilateral triangle | ||

c. | right triangle | ||

d. | isosceles triangle |

The distance between (

[Use the distance formula.]

AB =

[Replace (

=

[Simplify.]

BC =

[Replace (

=

[Simplify.]

CA =

[Replace (

=

[Simplify.]

AB = 12 units, BC = 15 units and CA = 9 units

BC

AB

AC

BC

[Since 225 = 144 + 81.]

ABC is a right triangle, since it satisfies the Pythagorean theorem.

Correct answer : (3)

28.

What type of triangle is PQR, if P(0, 12), Q(12, - 4) and R(12, 4)?

a. | isosceles triangle | ||

b. | right triangle | ||

c. | equilateral triangle | ||

d. | scalene triangle |

The distance between (

[Use the distance formula.]

PQ =

[Replace (

=

[Simplify.]

QR =

[Replace (

=

[Simplify.]

RP =

[Replace (

=

[Simplify.]

PQ = 20, QR = 8 and RP = 4

RP

RP

PQ ≠ QR ≠ RP. So PQR is a scalene triangle.

Correct answer : (4)

29.

The coordinates of B are formed by interchanging the coordinates of A. If the coordinates of A are (7, 11), then find the distance between A and B.

a. | 8 units | ||

b. | 4$\sqrt{2}$ units | ||

c. | 20 units | ||

d. | 5$\sqrt{2}$ units |

The coordinates of B are formed by interchanging the coordinates of A.

The coordinates of B are (11, 7).

Distance between the line-segment with endpoints (

Distance between A and B =

[Replace (

=

[Subtract.]

=

=

= 4

[Simplify.]

The distance between A and B is 4

Correct answer : (2)

30.

Find the distance between the center of the circle(C) and origin(0, 0) of the coordinate axes.

a. | 6$\sqrt{2}$units | ||

b. | 8$\sqrt{2}$units | ||

c. | 4$\sqrt{2}$units | ||

d. | 3$\sqrt{2}$ units |

Distance of a point (

[Write the distance formula.]

Distance between the center of the circle C and origin =

[Replace (

=

[Substitute the values.]

=

[Add.]

= 4

[Simplify.]

The distance between the center of the circle, C and origin of the coordinate axes is 4

Correct answer : (3)