Question

In the △PQR, PQ = 39 in, PR = 17 in, and the altitude PN = 15 in. Find QR.

Answer 1

Answer : The value of side QR is, 44 in.

Step-by-step explanation :

As we know that an altitude of a triangle is a segment from a vertex to the line containing its opposite side, and is perpendicular to that line.  

First we have to determine the side QN.

Using Pythagoras theorem in ΔPNQ :

$(Hypotenuse)^2=(Perpendicular)^2+(Base)^2$

$(PQ)^2=(PN)^2+(QN)^2$

Given:

Side PQ = 39

Side PN  = 15

Now put all the values in the above expression, we get the value of side QN.

$(39)^2=(15)^2+(QN)^2$

$QN=\sqrt{(39)^2-(15)^2}$

$QN=36$

Now we have to determine the side RN.

Using Pythagoras theorem in ΔPNR :

$(Hypotenuse)^2=(Perpendicular)^2+(Base)^2$

$(PR)^2=(PN)^2+(RN)^2$

Given:

Side PR = 17

Side PN  = 15

Now put all the values in the above expression, we get the value of side RN.

$(17)^2=(15)^2+(RN)^2$

$RN=\sqrt{(17)^2-(15)^2}$

$RN=8$

As,

Side QR = Side QN + Side RN

Side QR = 36 + 8

Side QR = 44

Thus, the value of side QR is, 44 in.

Answer 2

Answer:

So, this triangle PQR can be broken into two right triangles, PNQ and PNR, with legs PQ = 39, PN =15, and QN = ? and PR = 17, PN = 15, and NR =? respectively.

Let's solve for what is easier first:

Since we know that 5-15-17 is a Pythagorean triplet, we can infer that NR is 5....like I said earlier, it is a right triangle, so this guess holds true.

Here comes the interesting part:

Now, we have one part of QR, which is QN.

The other part can be solved by using the Pythagorean theorem.

It is (39^2-15^2)^(1/2)..which gives you 36, the square root of 1296, which happens to be the difference between the squares of 15 and 39.

SO, QR = QN + NR

5+36 = 41

QR = 41.

Hope this helps!