In the △PQR, PQ = 39 in, PR = 17 in, and the altitude PN = 15 in. Find QR.
can u guys show how to solve the problem(plz include answer)
i got 42
by using the pythagorean theorem i am not sure if i am correct but can u check it
PN^2 = 17^2-15^2 = 64
PN = 8
So, NQ = 39-8 = 31
QR^2 = 31^2+15^2 = 1186
QR = 34.438
PR=17cm
To find the length of QR in triangle PQR, you can use the Pythagorean theorem. Here are the steps to solve the problem:
1. Draw triangle PQR with sides PQ, PR, and the altitude PN.
2. From the given information, you know that PQ = 39 in, PR = 17 in, and PN = 15 in.
3. Since PN is an altitude, it divides the triangle into two right triangles: △PQN and △QRN.
4. In △PQN, you can use the Pythagorean theorem to find the length of QN. The Pythagorean theorem states that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse.
So, in △PQN:
PN^2 + QN^2 = PQ^2
15^2 + QN^2 = 39^2
225 + QN^2 = 1521
QN^2 = 1521 - 225
QN^2 = 1296
QN = √1296
QN = 36 in (rounded)
5. Now, in △QRN, you can use the Pythagorean theorem again to find the length of QR. Since QR is the hypotenuse in this triangle, you can apply the theorem as follows:
QR^2 = QN^2 + RN^2
QR^2 = 36^2 + PR^2
QR^2 = 1296 + 17^2
QR^2 = 1296 + 289
QR^2 = 1585
QR = √1585
QR ≈ 39.81 in (rounded)
So, the length of QR is approximately 39.81 inches.
To find the length of QR, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side (the hypotenuse).
In this case, triangle PQR is not specified as a right triangle. However, we can still use the concept of the altitude PN and the fact that it forms a right angle with QR to apply the Pythagorean Theorem.
Using the information given:
PQ = 39 in
PR = 17 in
PN = 15 in
First, let's find the length of QN by subtracting PN from PQ:
QN = PQ - PN = 39 in - 15 in = 24 in
Now, we can apply the Pythagorean Theorem to find the length of QR:
QR^2 = QN^2 + NR^2
Since triangle PQR is not specified as a right triangle, we don't know the value of NR^2. However, we can solve for NR^2 by using the fact that the altitude PN divides PR into two segments. Using the properties of similar triangles, we can set up the following proportion:
PN/PQ = NR/PR
Substituting the given values:
15 in / 39 in = NR / 17 in
Cross-multiplying, we get:
15 in * 17 in = NR * 39 in
Simplifying, we find:
NR = (15 in * 17 in) / 39 in = 6.5385 in (rounded to 4 decimal places)
Now, we can substitute the values of QN and NR into the Pythagorean Theorem equation:
QR^2 = (24 in)^2 + (6.5385 in)^2
Calculating this, we find:
QR^2 = 576 in^2 + 42.8039 in^2 = 618.8039 in^2
Taking the square root of both sides, we get:
QR = √(618.8039 in^2) ≈ 24.84 in (rounded to 2 decimal places)
Therefore, the length of QR is approximately 24.84 inches.