Worksheet: Applications of the Distance Formula

In this worksheet, we will practice using the law of the distance between two points to prove some of the geometrical shapes.

Q1:

What is the kind of triangle that the points 𝐴 ( 9 , 4 ) , 𝐵 ( 3 , 5 ) , and 𝐶 ( 6 , 1 ) form with respect to its angles?

  • Aright triangle
  • Bacute triangle
  • Cobtuse triangle

Q2:

Calculate the area of the parallelogram 𝑃 𝑄 𝑅 𝑆 , where the coordinates of its vertices are at 𝑃 ( 2 , 2 ) , 𝑄 ( 1 , 4 ) , 𝑅 ( 6 , 6 ) , and 𝑆 ( 3 , 0 ) .

Q3:

Find the area of the shape 𝐿 𝑀 𝑁 𝐻 below, where the coordinates of the points 𝐿 , 𝑀 , 𝑁 , and 𝐻 are ( 3 , 5 ) , ( 7 , 5 ) , ( 7 , 2 ) , and ( 3 , 2 ) , respectively, considering a unit length = 1 c m .

Q4:

Point ( 6 , 7 ) is on the circle with centre ( 7 , 1 ) . Decide whether point ( 8 , 9 ) is on, inside, or outside the circle.

  • Ainside the circle
  • Boutside the circle
  • Con the circle

Q5:

The coordinates of the points 𝐴 , 𝐵 , and 𝐶 are ( 𝐾 , 2 ) , ( 2 , 8 ) , and ( 9 , 6 ) respectively. Given that 𝐴 𝐵 = 𝐵 𝐶 , find all possible values of 𝐾 .

  • A 𝐾 = 3 or 𝐾 = 7
  • B 𝐾 = 3 or 𝐾 = 7
  • C 𝐾 = 3 or 𝐾 = 7
  • D 𝐾 = 3 or 𝐾 = 7

Q6:

An isosceles triangle has vertices , , and . Find the length of the line segment drawn from to which is perpendicular to .

  • A length units
  • B length units
  • C length units
  • D length units

Q7:

The points 𝐴 ( 7 , 7 ) , 𝐵 ( 7 , 6 ) , 𝐶 ( 6 , 6 ) , and 𝐷 ( 6 , 7 ) form the square 𝐴 𝐵 𝐶 𝐷 . What is its perimeter?

Q8:

In the grid, there are three points: 𝐷 , 𝐹 , and 𝐸 .

Find the fourth point which can be used to create a rectangle.

  • A ( 6 , 4 )
  • B ( 8 , 6 )
  • C ( 4 , 6 )
  • D ( 6 , 8 )
  • E ( 6 , 8 )

Find the length of that rectangle.

Q9:

A square drawn on a coordinate plane has corners at the following ordered pairs: ( 3 , 3 . 5 ) , ( 3 , 5 . 5 ) , and ( 5 , 3 . 5 ) . What is the ordered pair of the fourth corner?

  • A ( 5 . 5 , 5 )
  • B ( 5 , 7 . 5 )
  • C ( 3 . 5 , 5 )
  • D ( 5 , 5 . 5 )
  • E ( 7 , 5 . 5 )

Q10:

The points 𝐴 ( 0 , 1 ) , 𝐵 ( 𝑥 , 3 ) , and 𝐶 ( 5 , 2 ) are in a triangle with a right angle at 𝐵 . Determine the possible values of 𝑥 and the corresponding area of the triangle.

  • A 𝑥 = 4 , area = 4 or 𝑥 = 1 , area = 6 2 9 2
  • B 𝑥 = 4 , area = 4 4 1 or 𝑥 = 1 , area = 3 4 2
  • C 𝑥 = 4 , area = 4 1 7 or 𝑥 = 1 , area = 1 7 2 2
  • D 𝑥 = 4 , area = 4 or 𝑥 = 1 , area = 1 7 2

Q11:

Find the area of the isosceles triangle whose vertices are 𝐴 ( 6 , 3 ) , 𝐵 ( 2 , 9 ) , and 𝐶 ( 0 , 1 ) , with 𝐴 𝐵 = 𝐴 𝐶 .

Q12:

The line intersects the -axis at point and the -axis at point . Let be the origin. Find the area of triangle .

Q13:

Calculate, to two decimal places, the area of the triangle 𝑃 𝑄 𝑅 , where the coordinates of its vertices are at 𝑃 ( 5 , 1 , 2 ) , 𝑄 ( 4 , 4 , 3 ) , and 𝑅 ( 2 , 4 , 0 ) .

Q14:

Consider the following graphs of . We wish to compute the arc length between and , using line segments to approximate the curve.

At each refinement, we will subdivide our interval into twice as many subintervals as before. The figure shows the first two steps. The actual length of this arc, to 3 decimal places, is 4.277.

Find the approximate length using line segments, as in the figure. Give your answer to 3 decimal places.

Find the approximate length using line segments, as in the figure. Give your answer to 3 decimal places.

Q15:

Which of the following sets of points are collinear?

  • A ( 6 , 6 ) , ( 8 , 1 ) , ( 3 , 9 )
  • B ( 8 , 1 ) , ( 1 , 8 ) , ( 6 , 0 )
  • C ( 9 , 7 ) , ( 1 0 , 1 0 ) , ( 4 , 0 )
  • D ( 7 , 5 ) , ( 9 , 4 ) , ( 5 , 6 )

Q16:

A circle centred at ( 3 , 9 ) passes through point ( 1 , 1 ) . What is its radius? Give your answer to the nearest tenth.

Q17:

In a coordinate plane, a plot of the Kennedy Bridge that connects Louisville, KY, to Clarksville, IN, has a trapezoid middle section with vertices ( 3 , 7 ) , ( 4 , 3 ) , ( 3 , 3 ) , and ( 2 , 7 ) . Find the height of the trapezoid.

Q18:

A square has vertices at the points 𝐴 , 𝐵 , 𝐶 , and 𝐷 with coordinates ( 1 , 1 ) , ( 4 , 2 ) , ( 5 , 1 ) , and ( 2 , 2 ) respectively.

Work out the perimeter of the square 𝐴 𝐵 𝐶 𝐷 . Give your solution to two decimal places.

Work out the area of the square 𝐴 𝐵 𝐶 𝐷 .

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