In △PKQ, PK = KQ = 12, m∠P = 35º. Find PQ.

the triangle is isosceles, so m∠K = 110º

using the law of sines,
PQ/sin110º = 12/sin35º

To find the length of PQ in triangle PKQ, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and opposite angles A, B, and C, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In triangle PKQ, we have:

PK = KQ = 12 (given)
m∠P = 35º (given)

Let us label the length of PQ as x. Then, we have:

PK = KQ = 12
PQ = x
m∠P = 35º

We are interested in finding the length of PQ, so we can use the Law of Cosines with side PQ as side c and the angle opposite PQ, m∠P, as angle C.

c^2 = a^2 + b^2 - 2ab * cos(C)

PQ^2 = PK^2 + KQ^2 - 2 * PK * KQ * cos(m∠P)

Since PK = KQ = 12 and m∠P = 35º, we can substitute these values into the equation:

PQ^2 = 12^2 + 12^2 - 2 * 12 * 12 * cos(35º)

Now we can use a scientific calculator or math software to calculate the value of cos(35º) and simplify the equation further:

PQ^2 = 144 + 144 - 288 * cos(35º)

PQ^2 = 288 - 288 * cos(35º)

PQ^2 ≈ 288 - 235.59

PQ^2 ≈ 52.41

To find the value of PQ, we take the square root of both sides:

PQ ≈ √52.41

PQ ≈ 7.24

Therefore, in triangle PKQ, PQ is approximately equal to 7.24 units.