Congruence and similarity

Question

A triangular field ABC is such that AB = 90 m, BC = 70 m and AC = 85 m.

(i) Using a scale of 1 cm to 10 m, make a scale drawing of the field.

(ii) From your scale drawing, find the actual distance from B to the point K, where K is equidistant from A, B and C. Hint: Points on the perpendicular bisector of AB are equidistant from A and B.

Answer 1

You'll need to draw this out on your own. When you do it, you should get something that looks like this:

Draw the perpendicular bisectors to each side to find point K (where all three perpendicular bisectors intersect). Then measure segment BK. When you get your answer, convert it to actual "field" measurements by multiplying by 10 and using the unit "meters" (for instance, if you got 4.7 cm as your measurement for the length of segment BK, the actual field measurement would be 47 meters).

Answer 2

This sounds like you'll have to make a scaled model of the triangle and then find the circumcenter using a compass and ruler.

To make a scaled drawing, you will do the following.

ΔABC is your original triangle

ΔA'B'C' is your scaled down triangle

A'B' = AB x scale factor

A'B' = 90m x (1cm/10m) = 9cm

B'C' = BC x scale factor

B'C' = 70m x (1cm/10m) = 7cm

A'C' = 85m x (1cm/10m) = 8.5cm

You need to draw this using your ruler and compass.

I constructed it using the following

1) define a point A on your paper.

2) Draw a straight line from A to B that is 9cm long

3) From point A use a compass set to 8.5 cm and draw a circle

4) From point B use a compass and set it to 7cm and draw a circle

5) Those circles will intersect at two points. I chose 1 point and called it C. Draw a line from point A and Point B to this intersection point C

That is your constructed Triangle. with the appropriate side lengths. You can use your ruler to confirm the measurements.

Next you need to construct the circumcenter. The circumcenter is a point that is equidistant from all vertices of your triangle.

You will need to construct the perpendicular bisector of each side of the triangle.

Steps

1) Choose a side, I'll choose 9cm AB

2) Set your compass to a length that is greater than half of 9.5 (4.25) but less and 9cm .

3) Draw a circle from point A. Draw a circle from point B

4) the circles should intersect in two points. Draw a line connecting those two points. That line is the perpendicular bisector

Do this for all 3 sides.

The intersection of the 3 perpendicular bisectors is the circumcenter. This is the K' of your problem.

Now you have to use your ruler to measure the distance from A' , B', and C' to K'. That will give you your scaled distance. You will then have to scale that value up to get your actual distance.

When I did this on the geogebra website I measured the distance to k' to be 4.78cm

SO

Distance k' = k x scale factor

4.78 cm = k x (1cm/10m)

4.78cm (10m/1cm) = k'

47.8m = k

Here is the geogebra drawing

https://www.geogebra.org/calculator/f2shtunf

Hope this helps!!