What is the perimeter of triangle APQ in this geometry problem?

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In summary, the problem involves a circle with a radius of 9cm, with a tangent at points B, C, and R. The task is to find the perimeter of triangle APQ. Two methods were used to solve the problem, with the correct answer being 24 cm. However, the appropriateness of the methods is questioned and help is requested. The conversation also includes a discussion about the relationship between segments BP and PR, and RQ and CQ, as well as the similarity of triangles PRO and ORQ.
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timetraveller123
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Homework Statement


upload_2017-6-22_15-44-26.png

there is a circle with radius 9cm centred at o
segment oa is 15cm
ab, ca and pq are tangents to circle at points b,c and r respectively
find perimeter of triangle apq

Homework Equations

The Attempt at a Solution


i have solved the problem and got the right answer with two methods but i feel none of the methods is proper per say
method 1:
since the question is not imposing any restriction on the location of point r i assumed the perimeter is invariant with respect to r and hence let the segment pq fall on segment ca essentially the perimeter of two triangles is now just the two times the length of ac
ac is 12 cm by pythagoras hence the perimeter of triangle is 24 cm
a really crude method really don't think it is correct
method 2:
once with the assumption that the perimeter is invariant with respect to r
i shifted r right up to the point where da intersects the circle
then the triangle rpa is similar to triangle boa then rp is 9/2 and pa is 7.5 cm
then perimeter is once again 15 + 9 = 24 cm
however i feel my assumption is wrong and the question can be solved without the assumption so someone please help? thanks
 
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  • #2
vishnu 73 said:

Homework Statement


View attachment 205891
there is a circle with radius 9cm centred at o
segment oa is 15cm
ab, ca and pq are tangents to circle at points b,c and r respectively
find perimeter of triangle apq

Homework Equations

The Attempt at a Solution


i have solved the problem and got the right answer with two methods but i feel none of the methods is proper per say
method 1:
since the question is not imposing any restriction on the location of point r i assumed the perimeter is invariant with respect to r and hence let the segment pq fall on segment ca essentially the perimeter of two triangles is now just the two times the length of ac
ac is 12 cm by pythagoras hence the perimeter of triangle is 24 cm
a really crude method really don't think it is correct
method 2:
once with the assumption that the perimeter is invariant with respect to r
i shifted r right up to the point where da intersects the circle
then the triangle rpa is similar to triangle boa then rp is 9/2 and pa is 7.5 cm
then perimeter is once again 15 + 9 = 24 cm
however i feel my assumption is wrong and the question can be solved without the assumption so someone please help? thanks
upload_2017-6-22_14-37-57.png


How are the segments BP and PR related? The same with RQ and CQ?
 
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BP = BR and RQ = CQ
and i also note that triangle PRO is similar to ORQ
ok that is smart thanks
perimeter = PA + PQ + AQ = BA - BP + CA - CQ + PQ = BA + CA- PQ + PQ = 24 thanks
 
  • #4
vishnu 73 said:
BP = BR and RQ = CQ
and i also note that triangle PRO is similar to ORQ
Are they??
vishnu 73 said:
ok that is smart thanks
perimeter = PA + PQ + AQ = BA - BP + CA - CQ + PQ = BA + CA- PQ + PQ = 24 thanks
It was simple, was not it? :smile:
 
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  • #5
wait no they are not sorry my bad

yup it was simple
 

Related to What is the perimeter of triangle APQ in this geometry problem?

What is the definition of perimeter in geometry?

The perimeter in geometry refers to the distance around the outside of a two-dimensional shape. It is the sum of all the sides of the shape.

How do you find the perimeter of a shape?

To find the perimeter of a shape, you need to add up the length of all the sides. For example, if you have a rectangle with sides of 5 cm and 10 cm, the perimeter would be 5 cm + 5 cm + 10 cm + 10 cm = 30 cm.

What is the difference between perimeter and area?

Perimeter refers to the distance around the outside of a shape, while area refers to the amount of space inside the shape. Perimeter is measured in units of length, while area is measured in units squared.

Can the perimeter of a shape be negative?

No, the perimeter of a shape cannot be negative. Perimeter represents a physical distance, so it must be a positive value. If you encounter a negative perimeter, it is likely a mistake in calculation or measurement.

What are some real-life applications of perimeter in geometry?

Perimeter is used in various real-life situations, such as measuring the length of a fence, determining the amount of material needed to frame a picture, or calculating the distance around a track or sports field. It is also used in construction, architecture, and engineering to determine the length of walls and borders of structures.

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