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Lesson: Applications of the Distance Formula

Sample Question Videos

Worksheet • 18 Questions • 2 Videos

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 = βˆ’ 3 or 𝐾 = 7
  • B 𝐾 = 3 or 𝐾 = βˆ’ 7
  • C 𝐾 = βˆ’ 3 or 𝐾 = βˆ’ 7
  • D 𝐾 = 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 coordinates: , , and . What are the coordinates of the fourth corner?

  • A
  • B
  • C
  • D
  • E

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‘₯ = 1 , area = 1 7 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‘₯ = βˆ’ 1 , area = √ 6 2 9 2

Q7:

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

  • Aacute triangle
  • Bobtuse triangle
  • Cright triangle

Q8:

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

  • Aoutside the circle
  • Bon the circle
  • Cinside the circle

Q9:

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

  • Aoutside the circle
  • Binside the circle
  • Con 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 centred at ( βˆ’ 3 , βˆ’ 9 ) passes through point ( 1 , 1 ) . What is its radius? Give your answer to the nearest tenth.

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

Find the length of that rectangle.

Q16:

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.

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