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In this lesson, we will learn how to use the law of the distance between two points to prove some of the geometrical shapes.

Q1:

Find the area of the shape πΏ π π π» below, where the coordinates of the points πΏ , π , π , and π» are ( β 3 , β 5 ) , ( 7 , β 5 ) , ( 7 , 2 ) , and ( β 3 , 2 ) , respectively, considering a unit length = 1 c m .

Q2:

The coordinates of the points π΄ , π΅ , and πΆ are ( πΎ , β 2 ) , ( 2 , 8 ) , and ( β 9 , 6 ) respectively. Given that π΄ π΅ = π΅ πΆ , find all possible values of πΎ .

Q3:

Calculate the area of the parallelogram π π π π , where the coordinates of its vertices are at π ( β 2 , β 2 ) , π ( 1 , 4 ) , π ( 6 , 6 ) , and π ( 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?

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 ( β 3 , 7 ) , ( β 4 , β 3 ) , ( 3 , β 3 ) , and ( 2 , 7 ) . Find the height of the trapezoid.

Q6:

The points π΄ ( 0 , β 1 ) , π΅ ( π₯ , 3 ) , and πΆ ( 5 , 2 ) are in a triangle with a right angle at π΅ . Determine the possible values of π₯ and the corresponding area of the triangle.

Q7:

What is the kind of triangle that the points π΄ ( 9 , β 4 ) , π΅ ( 3 , 5 ) , and πΆ ( 6 , 1 ) form with respect to its angles?

Q8:

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

Q9:

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

Q10:

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

Q11:

Find the area of the isosceles triangle whose vertices are π΄ ( 6 , 3 ) , π΅ ( 2 , 9 ) , and πΆ ( 0 , β 1 ) , with π΄ π΅ = π΄ πΆ .

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

Q14:

The points π΄ ( 7 , β 7 ) , π΅ ( 7 , 6 ) , πΆ ( β 6 , 6 ) , and π· ( β 6 , β 7 ) form the square π΄ π΅ πΆ π· . What is its perimeter?

Q15:

In the grid, there are three points: π· , πΉ , and πΈ .

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

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.

Q17:

Calculate, to two decimal places, the area of the triangle β³ π π π , where the coordinates of its vertices are at π ( 5 , 1 , β 2 ) , π ( 4 , β 4 , 3 ) , and π ( 2 , 4 , 0 ) .

Q18:

Which of the following sets of points are collinear?

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