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In this lesson, we will learn how to find the distance between two points on the coordinate plane using the Pythagorean theorem.

Q1:

Find the distance between the point ( − 2 , 4 ) and the point of origin.

Q2:

Find the distance between the points 𝐴 and 𝐵 .

Q3:

Q4:

If the distance between the two points ( 𝑎 , 0 ) and ( − 𝑎 + 1 , 0 ) is 9, find all possible values of 𝑎 .

Q5:

If the distance between the two points ( 𝑎 , − 6 ) and ( 2 𝑎 − 4 , − 1 0 ) is 5, find all possible values of 𝑎 .

Q6:

If the distance between the two points ( 𝑎 , − 9 ) and ( 2 𝑎 + 2 , − 1 ) is 10, find all possible values of 𝑎 .

Q7:

How far is the butcher from the garage?

Q8:

On a map, coordinates are expressed in miles. The coordinates for San Francisco are ( 5 3 , 1 7 ) and those for Sacramento are ( 1 2 3 , 7 8 ) . Find the distance between the cities to the nearest mile.

Q9:

On a map, each coordinate represents a distance in miles from some fixed point. San Jose’s coordinates are (76, − 1 2 ), while those for San Francisco are (53, 17). Find the distance between San Jose and San Francisco to the nearest mile.

Q10:

The distance between ( 𝑎 , 5 ) and ( 1 , 1 ) is 5. What are the possible values of 𝑎 ?

Q11:

The points 𝐴 , 𝐵 , 𝐶 , and 𝐷 have coordinates ( 3 , 3 ) , ( 9 , 5 ) , ( − 2 , 8 ) , and ( 3 , − 1 ) . Which of the line segments 𝐴 𝐵 and 𝐶 𝐷 has the greatest length?

Q12:

Find the distance between the points ( 4 , 5 ) and ( 6 , − 2 ) .

Q13:

Find the distance between the points ( 2 , 3 ) and ( − 4 , − 4 ) .

Q14:

Find the distance between the two points in the figure below, giving your answer in radical form if necessary.

Q15:

Which of the following points is at a distance of 5 √ 2 from the origin?

Q16:

What is the distance between the point 𝐴 and the origin?

Q17:

Points 𝐴 and 𝐵 have coordinates 𝐴 ( 𝑥 , 𝑦 ) and 𝐵 ( 𝑥 , 𝑦 ) respectively. Point 𝐶 lies on 𝐴 𝐵 such that 𝐴 𝐶 and 𝐵 𝐶 are in the ratio of 𝑘 ∶ 𝑚 .

Write the quotient of the length of 𝐴 𝐶 by the length of 𝐴 𝐵 in terms of 𝑘 and 𝑚 .

Find an expression for the difference between the 𝑥 coordinates of 𝐴 and 𝐵 .

Find an expression for the difference between the 𝑦 coordinates of 𝐴 and 𝐵 .

Find an expression for the coordinates of 𝐶 .

Q18:

A small craft in Lake Ontario sends out a distress signal from the coordinates ( 4 9 , 6 4 ) . One rescue boat is at the coordinates ( 6 0 , 8 2 ) , while a Coast Guard boat one is at the coordinates ( 5 8 , 4 7 ) . Assuming both boat travel at the same rate, which one will get to the distressed boat first?

Q19:

Given 𝐴 ( 4 , 5 ) , 𝐵 ( 5 , 5 ) , and 𝐶 ( − 4 , − 7 ) , what is the perimeter of △ 𝐴 𝐵 𝐶 ?

Q20:

Consider the two points 𝐴 ( 𝑥 , 𝑦 ) and 𝐵 ( 𝑥 , 𝑦 ) .

Using the Pythagorean theorem, find an expression for the length of 𝐴 𝐵 .

Using the distance formula, find the distance between the points ( 3 , 4 ) and ( 5 , 6 ) . Give your answer as a surd in its simplest form.

Q21:

A triangle has vertices at the points 𝐴 ( 4 , 1 ) , 𝐵 ( 6 , 2 ) , 𝐶 ( 9 , 0 ) a n d .

Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form.

Is the triangle a right scalene triangle?

Q22:

The point 𝑃 is located on the line segment between point 𝐴 ( − 6 , 5 ) and point 𝐷 ( 9 , 8 ) . Given that the distance from 𝐴 to 𝑃 is twice the distance from 𝑃 to 𝐷 , find the coordinates of point 𝑃 .

Q23:

If 𝐴 ( 6 , 𝑥 ) and 𝐵 ( − 1 2 , − 9 ) , where 𝐴 𝐵 = 2 √ 1 4 5 length units, find all the possible values of 𝑥 .

Q24:

Two baseball posts were positioned at ( 9 , 3 ) and ( 6 , 9 ) . Find the distance between the posts giving the answer to one decimal place.

Q25:

Nabil is making a map of his local area measured in yards. The coffee shop is at ( − 5 , − 4 ) and the Italian restaurant is at ( 0 , 6 ) . Find the distance between the coffee shop and the Italian restaurant giving the answer to one decimal place.

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