Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Start Practicing

Worksheet: Applications of the Distance Formula

Q1:

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

Q2:

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

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

Q3:

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

Q4:

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 )

Q5:

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

Q6:

The points 𝐴 ( 0 , βˆ’ 1 ) , 𝐡 ( π‘₯ , 3 ) , and 𝐢 ( 5 , 2 ) are in a triangle with a right angle at 𝐡 . Determine the possible values of π‘₯ and the corresponding area of the triangle.

  • A π‘₯ = 4 , area = 4 or π‘₯ = βˆ’ 1 , area = √ 6 2 9 2
  • B π‘₯ = βˆ’ 4 , area = 4 √ 4 1 or π‘₯ = 1 , area = √ 3 4 2
  • C π‘₯ = 4 , area = 4 √ 1 7 or π‘₯ = 1 , area = 1 7 √ 2 2
  • D π‘₯ = 4 , area = 4 or π‘₯ = 1 , area = 1 7 2

Q7:

What is the kind of triangle that the points , , and form with respect to its angles?

  • Aright-angled triangle
  • Bacute-angled triangle
  • Cobtuse-angled triangle

Q8:

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

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

Q9:

Point ( 3 , βˆ’ 7 ) is on the circle with center ( 6 , 8 ) . Decide whether point ( βˆ’ 1 , βˆ’ 4 ) is on, inside, or outside the circle.

  • Aon the circle
  • Boutside the circle
  • Cinside the circle

Q10:

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

Q11:

Find the area of the isosceles triangle whose vertices are 𝐴 ( 6 , 3 ) , 𝐡 ( 2 , 9 ) , and 𝐢 ( 0 , βˆ’ 1 ) , with 𝐴 𝐡 = 𝐴 𝐢 .

Q12:

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

Q13:

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

  • A9.2
  • B116
  • C84
  • D10.8

Q14:

The points 𝐴 ( 7 , βˆ’ 7 ) , 𝐡 ( 7 , 6 ) , 𝐢 ( βˆ’ 6 , 6 ) , and 𝐷 ( βˆ’ 6 , βˆ’ 7 ) form the square 𝐴 𝐡 𝐢 𝐷 . What is its perimeter?

Q15:

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

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

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

Find the length of that rectangle.

Q16:

Consider the following graphs of 𝑦 = 1 π‘₯ + 4 ( π‘₯ βˆ’ 4 )   . We wish to compute the arc length between π‘₯ = 1 and π‘₯ = 3 , using line segments to approximate the curve.

At each refinement, we will subdivide our interval [ 1 , 3 ] 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 𝑛 = 1 line segments, as in the figure. Give your answer to 3 decimal places.

Find the approximate length using 𝑛 = 4 line segments, as in the figure. Give your answer to 3 decimal places.

Q17:

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

Q18:

Which of the following sets of points are collinear?

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