An isosceles triangle has two parallel sides and one non parallel side, the base. The angles are labelled q and r at the base and p at the point of the triangle. One parallel side is 44m in length, while the angle directly opposite from this side(angle q) is 86 degrees. Determine the perimeter of trainable PQR to the nearest tenth of a metre.
Thanks in advance
How can a triangle have two parallel sides?
I think you mean two congruent sides.
The angles are
Q=86
R=86
P=8
You know that q=r=44, so you just have to find p. Using the law of sines,
p/sin8 = 44/sin86
To determine the perimeter of triangle PQR, we need to calculate the lengths of the two equal sides (PQ and PR) using the given information, and then add them together with the length of the base (QR).
1. Let's start by finding the length of PQ and PR.
Since triangle PQR is an isosceles triangle, the two equal sides (PQ and PR) will have the same length.
2. We are given that one of the parallel sides is 44m in length.
Let's label this side as PR.
3. We also know that angle q (opposite the parallel side PR) is 86 degrees.
Since triangle PQR is isosceles, angle r (opposite the other parallel side) will also be 86 degrees.
4. To find the length of PQ, we can use the Law of Cosines since we have the lengths of two sides and the measure of one angle between them.
Let's use angle p and the side lengths PR and QR.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C),
where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.
Applying the Law of Cosines to triangle PQR, we have:
PQ^2 = PR^2 + QR^2 - 2 * PR * QR * cos(p).
Let's substitute the given values into the equation:
PQ^2 = 44^2 + QR^2 - 2 * 44 * QR * cos(p).
5. Since triangle PQR is an isosceles triangle, we know that angles q and r are congruent, so we can use the given measure of angle q (86 degrees) to find angle p.
Angle p = 180 - 2 * angle q.
Thus, angle p = 180 - 2 * 86 = 180 - 172 = 8 degrees.
6. Now we can calculate the length of PQ using the equation from step 4.
PQ^2 = 44^2 + QR^2 - 2 * 44 * QR * cos(8).
To find PQ, we take the square root of both sides:
PQ = √(44^2 + QR^2 - 2 * 44 * QR * cos(8)).
7. Once we know the length of PQ, we can calculate the perimeter of triangle PQR by adding the lengths of all three sides: PQ + QR + PR.
8. Substitute the calculated values of PQ and PR into the perimeter equation:
Perimeter = PQ + QR + PR.
Perimeter = √(44^2 + QR^2 - 2 * 44 * QR * cos(8)) + QR + 44.
9. Use algebraic manipulation to simplify the equation if needed.
10. Finally, calculate the value of the perimeter using a calculator and round to the nearest tenth of a meter to get the final answer.
I hope this helps! Let me know if you have any further questions.